Idea Transcript
UNIT 7
Transformations and Congruence CONTENTS
771A
COMMON CORE
MODULE 16
Tools of Geometry
G-CO.A.1 G-CO.A.1 G-CO.A.2 G-CO.C.9
Lesson 16.1 Lesson 16.2 Lesson 16.3 Lesson 16.4
Segment Length and Midpoints . . . . . . . . . . Angle Measures and Angle Bisectors . . . . . . . Representing and Describing Transformations Reasoning and Proof . . . . . . . . . . . . . . . . .
COMMON CORE
MODULE 17
Transformations and Symmetry
G-CO.A.4 G-CO.A.4 G-CO.A.4 G-CO.A.3
Lesson 17.1 Lesson 17.2 Lesson 17.3 Lesson 17.4
Translations . . . . . . . . Reflections. . . . . . . . . Rotations. . . . . . . . . . Investigating Symmetry
COMMON CORE
MODULE 18
Congruent Figures
G-CO.A.5 G-CO.B.6 G-CO.B.7
Lesson 18.1 Lesson 18.2 Lesson 18.3
Sequences of Transformations . . . . . . . . . . . . . . . . . . . . . . . 885 Proving Figures are Congruent Using Rigid Motions . . . . . . . . . 897 Corresponding Parts of Congruent Figures are Congruent . . . . . 909
Unit 7
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UNIT 7
Unit Pacing Guide 45-Minute Classes Module 16 DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 16.1
Lesson 16.2
Lesson 16.3
Lesson 16.3
Lesson 16.4
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 17.1
Lesson 17.2
Lesson 17.3
Lesson 17.3
Lesson 17.4
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
Lesson 18.1
Lesson 18.1
Lesson 18.2
Lesson 18.3
Module Review and Assessment Readiness
DAY 1
DAY 2
DAY 3
Lesson 16.1
Lesson 16.3
Lesson 16.4
DAY 6
Module Review and Assessment Readiness Module 17
DAY 6
Module Review and Assessment Readiness Module 18
DAY 6
Unit Review and Assessment Readiness
90-Minute Classes Module 16
Lesson 16.2
Module Review and Assessment Readiness
Module 17 DAY 1
DAY 2
DAY 3
Lesson 17.1
Lesson 17.3
Lesson 17.4
Lesson 17.2
Module Review and Assessment Readiness
Module 18 DAY 1
DAY 2
DAY 3
Lesson 18.1
Lesson 18.2
Module Review and Assessment Readiness
Lesson 18.3
Unit Review and Assessment Readiness
Unit 7
771B
Program Resources PLAN
ENGAGE AND EXPLORE
HMH Teacher App Access a full suite of teacher resources online and offline on a variety of devices. Plan present, and manage classes, assignments, and activities.
Real-World Videos Engage students with interesting and relevant applications of the mathematical content of each module.
Explore Activities Students interactively explore new concepts using a variety of tools and approaches.
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Professional Development Videos Authors Juli Dixon and Matt Larson model successful teaching practices and strategies in actual classroom settings. QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources. DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A
Teacher’s Edition Support students with point-of-use Questioning Strategies, teaching tips, resources for differentiated instruction, additional activities, and more. DONOT NOTEDIT--Changes EDIT--Changesmust mustbe bemade madethrough through"File "Fileinfo" info" DO CorrectionKey=NL-A;CA-A CorrectionKey=NL-A;CA-A
Name Name
Isosceles and Equilateral Triangles
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Class Class
_ _
Date Date
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22.2 Isosceles Isoscelesand andEquilateral Equilateral 22.2 Triangles Triangles
Common Core Math Standards
Investigating Isosceles Triangles INTEGRATE TECHNOLOGY
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View the Engage section online. Discuss the photo, explaining that the instrument is a sextant and that long ago it was used to measure the elevation of the sun and stars, allowing one’s position on Earth’s surface to be calculated. Then preview the Lesson Performance Task.
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QUESTIONING STRATEGIES
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Lesson2 2 Lesson
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Module2222 Module
1098 1098
Lesson2 2 Lesson
Date
22.2 Isosceles Triangles Essential
COMMON CORE
IN1_MNLESE389762_U8M22L21097 1097 IN1_MNLESE389762_U8M22L2
Question:
G-CO.C.10
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HARDCOVERPAGES PAGES10971110 10971110 HARDCOVER
PROFESSIONALDEVELOPMENT DEVELOPMENT PROFESSIONAL
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4/19/1412:10 12:10 PM 4/19/14 PM
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4/19/1412:10 12:10 PM 4/19/14 PM
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CONNECT VOCABULARY Ask a volunteer to define isosceles triangle and have students give real-world examples of them. If possible, show the class a baseball pennant or other flag in the shape of an isosceles triangle. Tell students they will be proving theorems about isosceles triangles and investigating their properties in this lesson.
Class
al and Equilater
Name
Proving the Isosceles Triangle Theorem and Its Converse
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In this activity, you will construct isosceles triangles and investigate other potential characteristics/properties of these special triangles.
EXPLAIN 1
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Module2222 Module
The angles that have the base as a side are the base angles.
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Checkstudents’ students’construtions. construtions. Check
BB
The side opposite the vertex angle is the base.
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© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company
Triangle33 Triangle
© Houghton Mifflin Harcourt Publishing Company © Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON PERFORMANCE TASK
Triangle22 Triangle
mm ∠∠ AA
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Explain to a partner what you can deduce about a triangle if it has two sides with the same length.
In an isosceles triangle, the angles opposite the congruent sides are congruent. In an equilateral triangle, all the sides and angles are congruent, and the measure of each angle is 60°.
Triangle11 Triangle
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Language Objective
Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles?
Legs Legs
Vertex angle
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Vertexangle angle Vertex
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Investigating Isosceles Triangles
An isosceles triangle is a triangle with at least two congruent sides.
Students have the option of completing the isosceles triangle activity either in the book or online.
Resource Locker
G-CO.C.10 Prove theorems about triangles.
Explore
CC
The student is expected to: COMMON CORE
COMMON CORE
EXPLORE
AA
BB
EssentialQuestion: Question:What Whatare arethe thespecial specialrelationships relationshipsamong amongangles anglesand andsides sidesininisosceles isosceles Essential andequilateral equilateraltriangles? triangles? and
Date
Essential Question: What are the special relationships among angles and sides in isosceles and equilateral triangles?
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ing Company
22.2
Class
22.2 Isosceles and Equilateral Triangles
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DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A
LESSON
Name
Base Base angles
PROFESSIONAL DEVELOPMENT
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C1
Lesson 19.2 Precision and Accuracy
Evaluate
1
Lesson XX.X ComparingLesson Linear, Exponential, and Quadratic Models 19.2 Precision and Accuracy
teacher Support
1
EXPLAIN Concept 1
Explain
The Personal Math Trainer provides online practice, homework, assessments, and intervention. Monitor student progress through reports and alerts. Create and customize assignments aligned to specific lessons or Common Core standards. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, Common Core standards, difficulty levels, and more. • Homework – Students can complete online homework with a wide variety of problem types, including the ability to enter expressions, equations, and graphs. Let the system automatically grade homework, so you can focus where your students need help the most! • Intervention – Let the Personal Math Trainer automatically prescribe a targeted, personalized intervention path for your students. 2
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4
Question 3 of 17
Concept 2
Determining Precision
ComPLEtINg thE SquArE wIth EXPrESSIoNS Avoid Common Errors Some students may not pay attention to whether b is positive or negative, since c is positive regardless of the sign of b. Have student change the sign of b in some problems and compare the factored forms of both expressions. questioning Strategies In a perfect square trinomial, is the last term always positive? Explain. es, a perfect square trinomial can be either (a + b)2 or (a – b)2 which can be factored as (a + b)2 = a 2 + 2ab = b 2 and (a – b)2 = a 2 + 2ab = b 2. In both cases the last term is positive. reflect 3. The sign of b has no effect on the sign of c because c = ( b __ 2 ) 2 and a nonzero number squared is always positive. Thus, c is always positive. c = ( b __ 2 ) 2 and a nonzero number c = ( b __ 2 ) 2 and a nonzero number
5
6
7
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9
10
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Solve the quadratic equation by factoring. 7x + 44x = 7x − 10
As you have seen, measurements are given to a certain precision. Therefore,
x=
the value reported does not necessarily represent the actual value of the measurement. For example, a measurement of 5 centimeters, which is
,
Check
given to the nearest whole unit, can actually range from 0.5 units below the reported value, 4.5 centimeters, up to, but not including, 0.5 units above it, 5.5 centimeters. The actual length, l, is within a range of possible values:
Save & Close
centimeters. Similarly, a length given to the nearest tenth can actually range from 0.05 units below the reported value up to, but not including, 0.05 units above it. So a length reported as 4.5 cm could actually be as low as 4.45 cm or as high as nearly 4.55 cm.
?
!
Turn It In
Elaborate
Look Back
Focus on Higher Order Thinking Raise the bar with homework and practice that incorporates higher-order thinking and mathematical practices in every lesson.
Differentiated Instruction Resources Support all learners with Differentiated Instruction Resources, including • Leveled Practice and Problem Solving • Reading Strategies • Success for English Learners • Challenge Calculate the minimum and maximum possible areas. Round your answer to
Assessment Readiness
the nearest square centimeters.
The width and length of a rectangle are 8 cm and 19.5 cm, respectively.
Prepare students for success on high stakes tests for Integrated Mathematics 1 with practice at every module and unit
Find the range of values for the actual length and width of the rectangle.
Minimum width =
7.5
cm and maximum width <
8.5 cm
My answer
Assessment Resources
Find the range of values for the actual length and width of the rectangle.
Minimum length =
19.45
cm and maximum length < 19.55
Name ________________________________________ Date __________________ Class __________________ LESSON
1-1
cm
Name ____________ __________________ __________ Date __________________ LESSON Class ____________ ______
Precision and Significant Digits
6-1
Success for English Learners
Linear Functions
Reteach
The graph of a linear The precision of a measurement is determined bythe therange smallest unit or Find of values for the actual length and width of the rectangle. function is a straig ht line. fraction of a unit used. Ax + By + C = 0 is the standard form for the equat ion of a linear functi • A, B, and C are on. Problem 1 Minimum Area = Minimum width × Minimum length real numbers. A and B are not both zero. • The variables x and y Choose the more precise measurement. = 7.5 cm × 19.45 cm have exponents of 1 are not multiplied together are not in denom 42.3 g is to the 42.27 g is to the inators, exponents or radical signs. nearest tenth. nearest Examples These are NOT hundredth. linear functions: 2+4=6 no variable x2 = 9 exponent on x ≥ 1 xy = 8 x and y multiplied 42.3 g or 42.27 g together 6 =3 Because a hundredth of a gram is smaller than a tenth of a gram, 42.27 g x in denominator x is more precise. 2y = 8 y in exponent Problem 2 In the above exercise, the location of the uncertainty in the linear y = 5 y in a square root measurements results in different amounts of uncertainty in the calculated Choose the more precise measurement: 36 inches or 3 feet. measurement. Explain how to fix this problem. Tell whether each function is linear or not. 1. 14 = 2 x 2. 3xy = 27 3. 14 = 28 4. 6x 2 = 12 x ____________
Reflect
____
________________
_______________
The graph of y = C is always a horiz ontal line. The graph always a vertical line. of x = C
_______________
is
Unit 7
Send to Notebook
_________________________________________________________________________________________
2. An object is weighed on three different scales. The results are shown Explore in the table. Which scale is the most precise? Explain your answer. Measurement
____________________________________________________________
• Tier 1, Tier 2, and Tier 3 Resources
Examples
1. When deciding which measurement is more precise, what should you Formula consider?
Scale
Tailor assessments and response to intervention to meet the needs of all your classes and students, including • Leveled Module Quizzes • Leveled Unit Tests • Unit Performance Tasks • Placement, Diagnostic, and Quarterly Benchmark Tests
Your Turn
y=1 T
x=2
y = −3
x=3
771D
Math Background Tools of Geometry
COMMON CORE
G-CO.A.1
LESSONS 16.1 and 16.2 The geometry concepts that are presented here are largely based on ideas set forth more than 2000 years ago by the Greek mathematician Euclid. His series of books, Elements (circa 300 B.C.E.), is the first known work in which a logical, deductive system of reasoning is used as a means of unifying all mathematical knowledge. Constructions with compass and straightedge date to antiquity. In fact, Euclid’s first three postulates describe how these tools may be used. The straightedge is used to draw a line through two points or to extend an existing line segment. In contrast to a ruler, the unmarked straightedge is never used for measuring distance. Although a compass is often thought of as a tool for making circles, its primary use in constructions is marking equal distances. Using these two seemingly primitive tools and working within the limits described above, one can construct virtually all of the fundamental figures of Euclidean geometry, including equilateral triangles, squares, and regular pentagons. An angle is defined as the figure that is formed by two rays with a common endpoint. Sometimes an angle is defined as the figure formed by two noncollinear rays with a common endpoint. This restriction eliminates straight angles and thus removes some of the ambiguity that can arise when straight angles are considered. (For example, either side of a straight angle could be considered the interior or exterior of the angle.) On the other hand, angles with measures greater than or equal to 180° are essential in trigonometry and other areas of higher mathematics. For this reason, straight angles are discussed here. Students should be aware, however, that in work with proofs, the term angle generally refers to angles formed by noncollinear rays, unless otherwise stated.
Reasoning and Proof
COMMON CORE
G-CO.C.9
LESSON 16.4 The essential doctrine of Elements is that when a certain set of fundamental ideas or understandings are assumed to be true, all other mathematical results can be logically derived and proved from these foundations.
771E
Unit 7
There are many forms for proofs, such as two-column proofs, flowchart proofs, and paragraph proofs. However, the main purpose of any proof, regardless of its format, is to present a logically sound argument. To that end, it is essential to point out to students that finding a proof and communicating a proof are two entirely different things. Students who see completed proofs should understand that these are models for how to present a finished logical argument; they are not models of the step-by-step thought process for finding a proof. The process of finding a proof is rarely a linear one. Instead, it is more often a matter of sorting through the pieces of a puzzle, looking for logical connections, stumbling into blind alleys, and returning to the starting point (sometimes more than once). Once this hit-and-miss process has been completed, the proof can be organized and communicated in any format. It is interesting to note that the first two-column proofs began appearing in textbooks around 1900. Since then, such proofs have often occupied a central role in geometry courses, and their prominence has led to misconceptions about their role, even among educators. According to a study that appeared in the Journal for Research in Mathematics Education, many pre-service teachers believe that an argument must be in two-column format in order to constitute a mathematical proof. Although the two-column format is practical because it reminds students that every statement must be supported with a reason, students should be reminded that the heart of any proof is the validity of the argument, not the format.
Transformations
COMMON CORE
G-CO.A.4
LESSONS 17.1 to 17.3 A transformation is a function that changes the position, size, or shape of a figure. Here, the emphasis is on transformations that are most closely linked to congruence and similarity: reflections, translations, rotations, and dilations. However, it is important to understand that there exist many other transformations. Perhaps the simplest transformation is the transformation that maps every point to itself. This is known as the identity transformation.
PROFESSIONAL DEVELOPMENT
Another simple transformation is the one that maps every point to the origin. An isometry is a transformation that preserves distance. This means that, under an isometry, the distance between any two points of the pre-image is the same as the distance between the corresponding points of the image. So, an isometry does not change the size or shape of a figure. Isometries are also called congruence mappings. In fact, congruence may be defined in terms of isometries as follows: Two figures are congruent if and only if there is an isometry that maps one figure to the other. In the plane, there are four types of isometries: reflections, translations, rotations, and glide reflections. The identity transformation, which maps every point to itself, may be considered a separate isometry or it may be considered a special case of a translation or rotation. It is natural to ask whether there are other isometries. For example, it seems intuitively obvious that a composition of isometries (one isometry followed by another) is also an isometry. Therefore, it makes sense to ask whether the composition of a translation and a rotation is a new type of isometry or whether the resulting transformation is equivalent to one of the four basic isometries. The somewhat surprising answer is that every isometry is indeed a reflection, translation, rotation, or glide reflection. As discussed above, every isometry is a reflection, translation, rotation, or glide reflection. Thus, the composition of any two of these isometries must be equivalent to one of the four basic isometries. The following table summarizes the results of all possible compositions. Composition of Isometries Refl.
Trans.
Rot.
Glide
Refl.
Trans. or Rot.
Glide
Glide
Trans. or Rot.
Trans.
Glide
Trans.
Rot.
Refl. or Glide
Rot.
Glide
Rot.
Trans. or Rot.
Glide
Glide
Trans. or Rot.
Refl. or Glide Glide
Symmetry
COMMON CORE
G-CO.A.3
LESSON 17.4 In general terms, an object, figure, or pattern has symmetry if a transformation can be performed on the object, figure, or pattern so that its image looks exactly like its preimage. If the relevant transformation is a reflection, the figure has line symmetry (or reflection symmetry). If the transformation is a rotation, the figure has rotational symmetry. It is also possible for a figure to have translation symmetry or glide-reflection symmetry, but these terms apply only to patterns that continue indefinitely, such as frieze patterns and tessellations.
Congruence and Corresponding Parts
COMMON CORE
G-CO.B.7
LESSONS 18.2 and 18.3 Two geometric figures are congruent if they are the same size and shape; in other words, if one of the figures can be moved so that it fits perfectly on top of the other figure. This is the intuitive idea behind the more rigorous mathematical definition of congruence: two figures are congruent if one can be transformed into the other by an isometry (that is, by a combination of translations, reflections, and rotations). For polygons, the definition of congruence can be stated in terms of corresponding sides and angles. In particular, two triangles are congruent if and only if the sides and angles can be matched up so that the corresponding sides are congruent and the corresponding angles are congruent. This definition of triangle congruence means that six correspondences must be checked in order to conclude that two triangles are congruent (three pairs of corresponding sides and three pairs of corresponding angles).
Trans. or Rot.
Unit 7
771F
UNIT
7
UNIT 7
Transformations and Congruence
MODULE
Transformations and Congruence
MATH IN CAREERS Unit Activity Preview
16
Tools of Geometry MODULE
17
Transformations and Symmetry MODULE
18
Congruent Figures
After completing this unit, students will complete a Math in Careers task by using given measurements in three dimensions to calculate distances. Critical skills include modeling real-world situations and using the distance formula.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Robert Garvey/Corbis
For more information about careers in mathematics as well as various mathematics appreciation topics, visit The American Mathematical Society at http://www.ams.org.
MATH IN CAREERS
Geomatics Surveyor A geomatics surveyor uses cutting-edge technology and math skills to make exact measurements of land, including distance and angle. Geomatics surveyors are important in the fields of construction, cartography, and oceanic engineering and exploration. If you’re interested in a career as a geomatics surveyor, you should study these mathematical subjects: • Algebra • Geometry • Trigonometry • Calculus Research other careers that require the use of spatial analysis to understand realworld scenarios. See the related Career Activity at the end of this unit. Unit 7
771
TRACKING YOUR LEARNING PROGRESSION
IN1_MNLESE389762_U7UO.indd 771
4/19/14 1:18 PM
Before
In this Unit
After
Students understand: • order of operations • using variables and expressions to represent situations • locating points in a coordinate plane • solving equations
Students will learn about: • segments and angles • reasoning and proof • translations, reflections, and rotations • symmetry • corresponding parts of congruent figures
Students will study: • properties of intersecting lines, parallel lines, and perpendicular lines • special segments of triangles • congruent triangles • geometric constructions
771
Unit 7
Reading Start -Up Visualize Vocabulary Use the ✔ words to complete the chart. You may put more than one word in each box. complementary supplementary angle angle
Angle
Description
acute
Angle whose measure is less than 90°
40°
50°
140°
Angle whose measure is greater than 90°
110°
None
70°
obtuse
Example
Reading Start Up
Vocabulary Review Words ✔ midpoint (punto medio) ✔ angle (ángulo) ✔ transformation (transformación) ✔ complementary angle (ángulo complementario) ✔ supplementary angle (ángulo suplementario) ✔ acute angle (ángulo agudo) ✔ obtuse angle (ángulo obtuso)
Have students complete the activities on this page by working alone or with others.
VISUALIZE VOCABULARY The description and example chart graphic helps students review vocabulary associated with angles. If time allows, brainstorm definitions and examples of transformations that students recall.
Preview Words angle bisector (bisectriz de un ángulo) vertex (vértice) collinear (colineales) postulate (postulado)
UNDERSTAND VOCABULARY Use the following explanations to help students learn the preview words. A postulate is a statement that is accepted without proof. An angle bisector is a ray that divides an angle into two equal angles. The endpoint of the ray lies at the vertex of the angle. Every point on the ray is collinear.
Understand Vocabulary Complete the sentences using the preview words. 1. 2.
A(n) angle bisector is a ray that divides an angle into two angles that both have the same measure. The common endpoint of two rays that form an angle is the vertex of the Points that lie on the same line are collinear .
Active Reading Booklet Before beginning each module, create a booklet to help you organize what you learn. As you study each lesson, draw the different graphical concepts that you learn and write their definitions.
© Houghton Mifflin Harcourt Publishing Company
angle.
3.
ACTIVE READING Students can use these reading and note-taking strategies to help them organize and understand the new concepts and vocabulary. Encourage students to speak up and ask for supplementary information to help them understand new vocabulary. Suggest that they include as much information as they need in their booklets to make included concepts clear.
ADDITIONAL RESOURCES Unit 7
Differentiated Instruction
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MODULE
16
Tools of Geometry
Tools of Geometry ESSENTIAL QUESTION: Answer: Just about any real-world situation involving shapes or the location of objects in space can be represented using the tools of geometry.
Essential Question: How can you use the tools of
geometry to solve real-world problems?
16 MODULE
LESSON 16.1
Segment Length and Midpoints LESSON 16.2
Angle Measures and Angle Bisectors
This version is for
Algebra 1 and PROFESSIONAL DEVELOPMENT Geometry only VIDEO
LESSON 16.3
Representing and Describing Transformations
Professional Development Video Author Juli Dixon models successful teaching practices in an actual high-school classroom.
my.hrw.com
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jochen Tack/Imagebroker/Corbis
Professional Development
LESSON 16.4
Reasoning and Proof
REAL WORLD VIDEO Check out how the tools of geometry can be used to solve real-world problems, like planning a park fountain’s location to be the same distance from the park’s three entrances.
MODULE PERFORMANCE TASK PREVIEW
How Far Is It? How does your cellphone know how far away the nearest restaurant is? In this module, you’ll explore how apps and search engines use GPS coordinates to calculate distances. So enter your present location and let’s find out!
Module 16
DIGITAL TEACHER EDITION IN1_MNLESE389762_U7M16MO 773
Access a full suite of teaching resources when and where you need them: • Access content online or offline • Customize lessons to share with your class • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most
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PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests.
4/19/14 12:56 PM
Are YOU Ready?
Are You Ready?
Complete these exercises to review skills you will need for this module.
ASSESS READINESS
Algebraic Representations of Transformations _
Shift y = √x horizontally 2 units to the right.
Example 1
(0, 0) to (2, 0)
Write the starting point and its transformation.
_
y – 0 = √x – 2
y=
Use the assessment on this page to determine if students need strategic or intensive intervention for the module’s prerequisite skills.
• Online Homework • Hints and Help • Extra Practice
Use the transformed point to write the equation.
_ √x – 2
Simplify.
Transform the equations. 1.
Shift y = 5x 3 units up.
2.
y = 5x + 3 3.
ASSESSMENT AND INTERVENTION
Stretch y = 5x vertically about the fixed x-axis by a factor of 2. y = 10x
_
Shift y = 5√x + 3 horizontally 2 units to the right and stretch by a factor of 3. (Stretch vertically about the fixed y = 3 line.) _ y = 15√ x – 2 + 3
Angle Relationships
3 2 1
Find the angle complementary to the given angle, 75°.
Example 2
x + 75° = 90°
Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student’s individual needs!
Write as an equation.
x = 90° - 75°
Solve for x.
x = 15°
Find the complementary angle. 4.
70°
5.
35°
55°
6.
67°
23°
8.
65°
115°
9.
34°
146°
© Houghton Mifflin Harcourt Publishing Company
20°
Find the supplementary angle. 7.
80°
100°
Distance and Midpoint Formulas Example 3
Find the distance between (2, 3) and (5, 7). ―――――― √(5 – 2) 2 + (7 – 3) 2 Apply the distance formula. _
= √ 9 + 16
Simplify each square.
=5
Add and find the square root.
Find each distance and midpoint for the given points. 10. The points (6, 14) and (1, 2) 11. The points (4, 6) and (19, 14) Module 16
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Distance Distance
13
Midpoint
17
Midpoint
TIER 1, TIER 2, TIER 3 SKILLS
ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: • Tier 2 Skill Pre-Tests for each Module • Tier 2 Skill Post-Tests for each skill
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Response to Intervention
Tier 1 Lesson Intervention Worksheets
Tier 2 Strategic Intervention Skills Intervention Worksheets
Tier 3 Intensive Intervention Worksheets available online
Reteach 16.1 Reteach 16.2 Reteach 16.3 Reteach 16.4
33 Algebraic Representations of Transformations 34 Angle Relationships 38 Distance and Midpoint Formulas
Building Block Skills 7, 10, 11, 15, 16, 27, 38, 45, 46, 51, 53, 55, 56, 66, 69, 70, 95, 98, 100, 102,
Differentiated Instruction
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Challenge worksheets Extend the Math Lesson Activities in TE
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LESSON
16.1
Name
Segment Length and Midpoints
Essential Question: How do you draw a segment and measure its length?
Resource Locker
Explore
The student is expected to:
The most basic figures in geometry are undefined terms, which cannot be defined using other figures. The terms point, line, and plane are undefined terms. Although they do not have formal definitions, they can be described as shown in the table.
Mathematical Practices MP.5 Using Tools
Language Objective
Undefined Terms
Work with a small group to match pictures to “geometry term cards.”
Possible answer: You can use a compass and straightedge to draw a segment and use a ruler to measure it. Or, you can connect two points on a coordinate plane to form a segment and use the Distance Formula to find its length.
PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and the fact that the fence must be big enough to enclose the ride in all of its possible configurations. Then preview the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Marco Vacca/age fotostock
ENGAGE Essential Question: How do you draw a segment and measure its length?
Exploring Basic Geometric Terms
In geometry, some of the names of figures and other terms will already be familiar from everyday life. For example, a ray like a beam of light from a spotlight is both a familiar word and a geometric figure with a mathematical definition.
G-CO.A.1
Know precise definitions of ... line segment, based on the undefined notions of ... distance along a line, … . Also G-CO.D.12, G-GPE.B.4 COMMON CORE
Date
16.1 Segment Length and Midpoints
Common Core Math Standards COMMON CORE
Class
Term
Geometric Figure
A point is a specific location. It has no dimension and is represented by a dot.
P
A line is a connected straight path. It has no thickness and it continues forever in both directions. A plane is a flat surface. It has no thickness and it extends forever in all directions.
point P ℓ
B
A
Ways to Name the Figure
X
line ℓ, line AB, line BA, ‹ › − ‹ › − AB, or BA
Z
plane or plane XYZ
Y
In geometry, the word between is another undefined term, but its meaning is understood from its use in everyday language. You can use undefined terms as building blocks to write definitions for defined terms, as shown in the table.
Defined Terms Term
Geometric Figure
A line segment (or segment) is a portion of a line consisting of two points (called endpoints) and all points between them.
C
A ray is a portion of a line that starts at a point (the endpoint) and continues forever in one direction.
Module 16
Ways to Name the Figure segment _ _CD, segment DC, CD, or DC
D
→ ‾ ray PQ or PQ
P Q
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Undefined
Geometric
Terms Term
Figure
Name the
Watch for the hardcover student edition page numbers for this lesson.
Figure
point P
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X
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You can use points to sketch lines, segments, rays, and planes.
A
Draw two points J and K. Then draw a line through them. (Remember that a line shows arrows at both ends.)
B
EXPLORE
Draw two points J and K again. This time, draw the line segment with endpoints J and K.
Exploring Basic Geometric Terms
J
J
K
INTEGRATE TECHNOLOGY
K
C
Draw a point K again and draw a ray from endpoint K. Plot a point J along the ray.
D
Draw three points J, K, and M so that they are not all on the same line. Then draw the plane that contains the three points. (You might also put a script letter such as on your plane.)
J
J M
K
E
Line Segment
Connect the words collinear and coplanar to the prefix co-. Let students know that co- usually means together or joint. For example, coauthors author a book together. Collinear means that points are on the same line, coplanar means “together on the same plane.”
K
Plane
points J, K, and M
undefined term/defined term
‹ › ‹ › − − JK (or KJ )
QUESTIONING STRATEGIES
undefined term/defined term
_ _ JK or KJ
How are drawing and naming lines, rays, and line segments similar and how are they different? All of the figures can be named by two points. The line segment connects the two points, the ray extends beyond one of the named points with an arrow at the extended end, and the line extends beyond both named points, with arrows at each end.
undefined term/defined term
→ ‾ KJ
undefined term/defined term
plane JKM (or plane )
undefined term/defined term
Reflect
→ → ‾ be the same ray as KJ ‾ ? Why or why not? In Step C, would JK No. The rays would have different endpoints and continue in opposite directions. In Step D, when you name a plane using 3 letters, does the order of the letters matter? No. Using 3 letters, the plane in Step D can be named plane JKM, plane JMK, plane KJM,
© Houghton Mifflin Harcourt Publishing Company
Ray
2.
CONNECT VOCABULARY
Give a name for each of the figures you drew. Then use a circle to choose whether the type of figure is an undefined term or a defined term. Point
1.
Geometry programs and other software contain tools to measure segment lengths and distances.
Does the order of the two points matter when naming a line segment, a ray, or a line? Explain. The order does not matter when naming a line or a line segment but it does matter when naming a ray. Each named ray has a different endpoint and continues forever in opposite directions.
plane KMJ, plane MJK, or plane MKJ. 3.
‹ › ‹ › − − Discussion If PQ and RS are different names for the same line, what must be true about points P, Q, R, and S? The four points all lie on a common line.
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Math Background
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Students have worked with geometric terms and figures since the elementary grades. This course revisits many ideas that may be familiar to students, but does so in a systematic way in order to build a deductive system. The Distance and Midpoint Formulas are key tools of coordinate geometry. Students will write coordinate proofs later in this course and they will find that these two formulas, along with facts about the slopes of parallel and perpendicular lines, are enough to prove a wide range of theorems.
Segment Length and Midpoints
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Explain 1
EXPLAIN 1
The distance along a line is undefined until a unit distance, such as 1 inch or 1 centimeter, is chosen. You can use a ruler to find the distance between two points on a line. The distance is the absolute value of the difference of the numbers on the ruler that 0cm correspond to the two points. This distance is the length of the segment determined by the points. _ In the figure, the length of RS, written RS (or SR), is the distance between R and S.
Constructing a Copy of a Line Segment CONNECT VOCABULARY
RS = ⎜4 - 1⎟ = ⎜3⎟ = 3 cm
The definition of the distance used to measure the length of a line segment is called the Ruler Postulate.
or
S R
1
2
3
4
5
SR = ⎜1 - 4⎟ = ⎜-3⎟ = 3 cm
Points that lie in the same plane are coplanar. Lines that lie in the same plane but do not intersect are parallel. Points that lie on the same line are collinear. The Segment Addition Postulate is a statement about collinear points. A postulate is a statement that is accepted as true without proof. Like undefined terms, postulates are building blocks of geometry.
QUESTIONING STRATEGIES
Postulate 1: Segment Addition Postulate Let A, B, and C be collinear points. If B is between A and C, then AB + BC = AC.
Why is the first step in constructing a copy of a line segment to draw a line segment with an endpoint? It provides a place to set the compass point and draw the arc that shows the length of the copied segment.
A
B
C
A construction is a geometric drawing that produces an accurate representation without using numbers or measures. One type of construction uses only a compass and straightedge. You can construct a line segment whose length is equal to that of a given segment using these tools along with the Segment Addition Postulate. Example 1
AVOID COMMON ERRORS
Use a compass and straightedge to construct a segment whose length is AB + CD.
A
B D
C
Step 1 Use the straightedge to draw a long line segment. Label an endpoint X. (See the art drawn in Step 4.) © Houghton Mifflin Harcourt Publishing Company
Remind students to set the compass point on the endpoint of the line segment and to be careful not to change the distance setting before drawing the arc to copy a segment.
Constructing a Copy of a Line Segment
Step 2 To copy segment AB, open the compass to the distance AB.
A
B
Step 3 Place the compass point on X, and draw an arc. Label the point Y where the arc and the segment intersect. Step 4 To copy segment CD, open the compass to the distance CD. Place the compass point on Y, and draw an arc. Label the point Z where this second arc and the segment intersect.
X
Y
Z
_ XZ is the required segment. Module 16
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Peer-to-Peer Activity Have students work in pairs and use a compass and straightedge to construct a fair ruler. Tell the students to draw a straight line, set the compass for the unit, and after marking an endpoint as 0, construct and label the ruler. Have students compare their rulers. Discuss which units are fair rulers and why they are, or are not. If time permits, discuss which rulers would be most appropriate to measure the lengths of different objects.
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B
B
A C
D
Step 1 Use the straightedge to draw a long line segment. Label an endpoint X. Step 2 To copy segment AB, open the compass to the distance AB. Step 3 Place the compass point on X, and draw an arc. Label the point Y where the arc and the segment intersect. Step 4 To copy segment CD, open the compass to the distance CD. Place the compass point on Y, and draw an arc. Label the point Z where this second arc and the segment intersect. _ XZ is the required segment. X
Y
Z
Reflect
4.
Discussion Look at the line and ruler above Example 1. Why does it not matter whether you find the distance from R to S or the distance from S to R? The formula to find the distance between the two points involves taking the absolute value of the difference between the two coordinates R and S, so the distance is always positive; the order of the coordinates does not matter. From R to S or from S to R, the coordinates are always 3 units apart.
5.
_ _ In Part B, how can you check _length of YZ is the same as the length of CD? _ that the Use a ruler to measure YZ and CD to see if the lengths are the same.
Your Turn © Houghton Mifflin Harcourt Publishing Company
6.
Use a ruler to draw a segment PQ that is 2 inches _ long. Then use your compass and straightedge to construct a segment MN with the same length as PQ. P
Q
M
Explain 2
N
Using the Distance Formula on the Coordinate Plane
The Pythagorean Theorem states that a 2 + b 2 = c2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. You can use the Distance Formula to apply the Pythagorean Theorem to find the distance between points on the coordinate plane.
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Communicating Math
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―――
Before introducing the Distance Formula, ask students to explain why 3 2 + 4 2 ≠ 3 + 4. You must first square each number and then take the square root of the sum. The answer is 5, not 7.
Visual Cues Have students estimate the midpoint or distance of a line segment plotted in the coordinate plane. Then have them use a calculator and the appropriate formula to support their answers.
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EXPLAIN 2
The Distance Formula The distance between two points (x 1, y 1) and (x 2, y 2) on the coordinate
―――――――― plane is √(x - x ) + (y - y ) .
Using the Distance Formula on the Coordinate Plane
2
AVOID COMMON ERRORS
Example 2
Students may confuse the coordinates when using the Distance Formula. Have them label the coordinates of any two points they are given as (x 1, y 1) and (x 2, y 2) before substituting the numbers into the Distance Formula.
2
2
1
(x2, y2)
2
(x1, y1)
x
Determine whether the given segments have the same length. Justify your answer.
y
4
A
C
E
B
-4
0
x H
4 D
F G
QUESTIONING STRATEGIES
-4
_ _ AB and CD Write the coordinates of the endpoints. _ Find the length of AB. Simplify the expression. _ Find the length of CD. Simplify the expression.
―
A(-4, 4), B(1, 2), C(2, 3), D(4, −2)
―――――――― + (2 - 4) ―――― = √5 + (-2) = √― 29 ―――――――― CD = √(4 - 2) + (-2 - 3) ―――― 29 = √2 + (-5) = √― AB =
√(1 - (-4))
2
2
2
2
2
2
2
2
_ _ So, AB = CD = √29 . Therefore, AB and CD have the same length. © Houghton Mifflin Harcourt Publishing Company
How can you use the Pythagorean Theorem to find the distance between two points in the plane if you forget the Distance Formula? Use the endpoints to draw a right triangle with a vertical leg and a horizontal leg. The hypotenuse is the line connecting the points. Find the length of each leg and then use the Pythagorean Theorem to find the length of the hypotenuse.
1
y
_ _ EF and GH Write the coordinates of the endpoints. _ Find the length of EF. Simplify the expression. _ Find the length of GH. Simplify the expression.
(
)
E(-3, 2), F -2 , -3 , G(-2, -4), H ――――――――――― 2 2
EF =
√(
) (
)
(2
, 0
)
-2 - (-3) + -3 - 2
___
= √( 1 ) + ( -5 ) = √ 26 2
2
――
―――――――――――― 2 2
GH =
√(
) (0
2 - (-2) +
___
- (-4)
= √( 4 ) + ( 4 ) = √ 32 2
2
――
)
_ _ So, EF ≠ GH . Therefore, EF and GH do not have the same length. Module 16
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Connect Vocabulary Have students look up the word between in the dictionary and compare that definition with the mathematical definition. Emphasize that the mathematical definition of between includes collinearity.
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EXPLAIN 3
Reflect
7.
Consider how the Distance Formula is related to the Pythagorean
4
Theorem. To use the Distance Formula to____ find the distance from 2 2 U(−3, −1) to V(3, 4), you write UV = √ (3 - (-3)) + (4 - (-1)) .
Explain how (3 - (−3)) in the Distance Formula is related to a in the
y
V
Finding a Midpoint x
Pythagorean Theorem and how (4 - (−1)) in the Distance Formula is
-4 U
related to b in the Pythagorean Theorem.
0
CONNECT VOCABULARY
4
Explain that bi- is a prefix meaning two and sect means to cut, as into sections. A segment bisector divides the segment into two equal parts.
-4
The Pythagorean Theorem states that a 2 + b 2 = c 2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. Applying this to the right ――― triangle in the figure, UV = c = √a 2 + b 2 , where a is the length of the horizontal leg of the
triangle, or (3 - (−3)), and b is the length of the vertical leg of the triangle, or (4 - (−1)). Your Turn
8.
_ _ Determine whether JK and LM have the same length. Justify your answer.
J(−4, 4), K(−2, 1), L(−1, −4), M(2, −2) ―――――――― 2 ― 2 JK = (-2 - (-4)) + (1 - 4) = √13 ―――――――――― 2 2 ― LM = (2 - (-1)) + (-2 - (-4)) = √13 ― ― ― So, JK = LM = √13 . Therefore, JK and LM have the same length.
√ √
4
R K
-4
0
P
y
S x
Q M
4
L
© Houghton Mifflin Harcourt Publishing Company
Explain 3
J
Finding a Midpoint
The midpoint of a line segment is the point that divides the segment into two segments that have the same length. A line, ray, or other figure that passes through the midpoint of a segment is a segment bisector. _ In the figure, _ the tick marks show that PM = MQ. Therefore, M is the midpoint of PQ and line ℓ bisects PQ. ℓ P
M
Q
You can use paper folding as a method to construct a bisector of a given segment and locate the midpoint of the segment.
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AVOID COMMON ERRORS
Use paper folding to construct a bisector of each segment.
Example 3
Some students may have difficulty aligning the points so that one is on top of the other. Have students enlarge and darken the endpoints so they are easier to locate. Students can fold either inward or outward as long as the points are put together.
B
B B
A
A
A
Step 2 Fold the paper so that point B is on top of point A.
Step 1 Use a compass and straightedge _ to copy AB on a piece of paper.
B
QUESTIONING STRATEGIES
AB
A
When you use paper folding to bisect a line segment, why do you fold the paper so that the endpoints of the line segment are on top of each other? Placing one endpoint on top of the other creates a mirror image of the parts, with the crease corresponding to the midpoint and mirror line.
A
Step 3 Open the paper. Label the point where the crease intersects the segment as point M. B B
A
M
M
A
A
_ _ Point M is the midpoint of AB and the crease is a bisector of AB.
© Houghton Mifflin Harcourt Publishing Company
B _ M straightedge to copy JK on a piece of paper. Step 1 Use a compass and
Step 2 AFold the paper so that point K is on top of point
J
J
.
Step 3 Open the paper. Label the point where the crease intersects the segment as point N. Point N is the midpoint
―
of JK and the crease is a
bisector
N K
―
of JK.
Step 4 Make a sketch of your paper folding construction or attach your folded piece of paper. Reflect
9.
Explain how you could use paper folding to divide a line segment into four segments of equal length. Use paper folding to construct the midpoint of the segment. Then use the same methods
to construct the midpoint of each of the two new segments. The three midpoints divide the given segment into four segments of equal length.
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Your Turn
EXPLAIN 4
10. Explain how to use a ruler to check your construction in Part B. Measure each of the segments formed by the bisector. The two segments should each have
Finding Midpoints on the Coordinate Plane
a length that is half as long as the given segment.
Explain 4
Finding Midpoints on the Coordinate Plane
AVOID COMMON ERRORS
You can use the Midpoint Formula to find the midpoint of a segment on the coordinate plane.
To avoid computation errors, caution students to pay attention to the signs of the endpoint coordinates when they are finding the midpoint. Students may benefit from plotting segments in the plane before using the formula to find the midpoint.
The Midpoint Formula _ The midpoint M of AB with endpoints A(x 1, y 1) and B(x 2, y 2) y1 + y2 x1 + x2 _ , is given by M _ . 2 2
(
Example 4
y
)
B(x2, y2) M A(x1, y1)
( x +2 x , y +2 y ) 1
2
1
2
x
Show that each statement is true.
QUESTIONING STRATEGIES
_ _ If PQ has endpoints P(-4, 1) and Q(2, −3), then the midpoint M of PQ lies in Quadrant III.
(
―
Does it matter which point is represented by (x 1, y 1) ? Explain. No, the midpoint coordinates are the same due to the Commutative Property of Addition.
)
-4 + 2 1 + (-3) M _, _ = M(-1, -1) 2 2
Use the Midpoint Formula to find the midpoint of PQ. Substitute the coordinates, then simplify.
So M lies in Quadrant III, since the x- and y-coordinates are both negative.
_ _ If RS has endpoints R(3, 5) and S(−3, −1), then the midpoint M of RS lies on the y-axis.
)
(0
Substitute the coordinates, then simplify.
, 2
)
So M lies on the y-axis, since the x-coordinate is 0 . Your Turn
Show that each statement is true. _ 11. If AB has endpoints A(6,_ −3) and B(-6, 3), then the midpoint M of AB is the origin.
_ 12. If JK has endpoints J(7, 0) and K(−5, −4), _ then the midpoint M of JK lies in Quadrant IV.
6 + (-6) -3 + 3 M _, _ = M(0, 0) 2 2
7 + (-5) 0 + (-4) M _, _ = M(1, -2) 2 2
(
)
(
So M is the origin, since the xand y-coordinates are both 0.
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)
© Houghton Mifflin Harcourt Publishing Company
(
3 + -3 5 + -1 M _, _ = M 2 2
_ Use the Midpoint Formula to find the midpoint of RS.
How could you use the Distance Formula to check the location of a midpoint? Check that the distance from one endpoint to the midpoint equals the distance from the other endpoint to the midpoint.
So M lies in Quadrant IV, since the x-coordinate is positive and the y-coordinate is negative. 782
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Elaborate
ELABORATE
13. Explain why the Distance Formula is not needed to find the distance between two points that lie on a horizontal or vertical line. If two points lie on a horizontal or vertical line, they share a common x-coordinate or
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Discuss with students why they are applying
y-coordinate. To find the distance between the points, you just need to find the positive difference of the other coordinates. 14. When you use the Distance Formula, does the order in which you subtract the x- and y-coordinates matter? Explain. No; (x 1 - x 2)2 = (x 2 - x 1) 2 and (y 1 - y 2) 2 = (y 2 - y 1) 2.
the Ruler Postulate (“The distance between any two points is equal to the absolute value of the difference of their coordinates”) when they find the length of a vertical or a horizontal segment on a coordinate plane.
15. When you use the Midpoint Formula, can you take either point as (x 1, y 1) or (x 2, y 2)? Why or why not? y1 + y2 y2 + y1 x1 + x2 x2 + x1 Yes; = and = . 2 2 2 2
_ _
16. Essential Question Check-In What is the difference between finding the length of a segment that is drawn on a sheet of blank paper and a segment that is drawn on a coordinate plane? Possible answer: You use a ruler to find the length of a segment drawn on a sheet of blank
AVOID COMMON ERRORS
QUESTIONING STRATEGIES Can you use the Distance Formula to find the length of a vertical or horizontal line segment? Explain. Yes, either the horizontal or vertical distance reduces to 0, so it is the same as finding the positive difference of using the other coordinate.
SUMMARIZE THE LESSON If you know the endpoints of a line segment, how can you find the length of the line segment and its midpoint? Use the Distance Formula to find the length of the segment and the Midpoint Formula to find the coordinates of its midpoint.
paper and the Distance Formula to find the length of a segment on a coordinate plane.
Evaluate: Homework and Practice © Houghton Mifflin Harcourt Publishing Company • Image Credits: (r) ©Ocean/ Corbis
Remind students that they can draw a quick sketch to help them recognize when a line is horizontal, vertical, or oblique to help them select the appropriate method to find the length of a line given its coordinates.
Write the term that is suggested by each figure or description. Then state whether the term is an undefined term or a defined term. 1.
Lesson 16.1
• Online Homework • Hints and Help • Extra Practice
2.
line segment; defined term 3.
M
point; undefined term 4.
2
L
y x
-2
ray; defined term
0
2
-2
plane or line; undefined term Module 16
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Use a compass and straightedge to construct a segment whose length is AB + CD. 5.
6.
A
B
C
D
X
C
Y
D
X
Z
EVALUATE
B
A
Y
Z
Copy each segment onto a sheet of paper. Then use paper folding to construct a bisector of the segment. Check students’ constructions. 7.
L
8.
A
ASSIGNMENT GUIDE
B
K
Determine whether the given segments have the same length. Justify your answer. _ _ 9. AB and BC A(-4, 2), B(1, 4), C(2, -1)
―――――――― 29 √(1 - (-4)) + (4 - 2) = √― ―――――――― BC = √(2 - 1) + (-1 - (4)) = √― 26 _ _ 2
AB =
4
D
A F
2
-4
2
2
y B
AB ≠ BC, so AB and BC do not have the same length. _ _ 10. EF and GH
0
E
G
E(-4, -3), F(-1, 1), G(-2, -3), H(3, -3)
x 4
C H
-4
―――――――――― EF = √(-1 - (-4)) + (1 - (-3)) = 5 ―――――――――― GH = √(3 - (-2)) + (-3 - (-3)) = 5 _ _ 2
2
2
2
B(1, 4), C(2, -1), E(-4, -3), F(-1, 1)
A(-4, 2), B(1, 4), C(2, -1), D(4, 4)
―――――――― AB = √(1 - (-4)) + (4 - 2) = √― 29 ―――――――― BC = √(4 - 2) + (4 - (-1)) = √― 29 _ _ ― So, AB = CD = √29 . Therefore, AB and CD 2
2
EF =
2
Show that each statement is true. _ 13. If DE has endpoints D( −1, _ 6 ) and E(3, −2), then the midpoint M of DE lies in Quadrant I.
2
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–4
1 Recall of Information
MP.6 Precision
5–8
2 Skills/Concepts
MP.5 Using Tools
9–12
2 Skills/Concepts
MP.2 Reasoning
13–16
2 Skills/Concepts
MP.2 Reasoning
17–18
1 Recall of Information
MP.3 Logic
19–20
2 Skills/Concepts
MP.4 Modeling
Exercises 1–4
Example 1 Constructing a Copy of a Line Segment
Exercises 5–6
Example 2 Using the Distance Formula on the Coordinate Plane
Exercises 9–12
Example 3 Finding a Midpoint
Exercises 7–8
Example 4 Finding Midpoints on the Coordinate Plane
Exercises 13–16
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 As one student copies a line segment, have
Lesson 1
COMMON CORE
Explore Exploring Basic Geometric Terms
Students sometimes forget to place a symbol above letters used to name lines, segments, and rays. Remind them that two letters without a symbol represent a length.
So M lies on the x-axis, since the y-coordinate is 0. 784
Practice
COMMUNICATING MATH
(_ _)
2 2 So M lies in Quadrant I, since the xand y-coordinates are both positive.
Exercise
2
_ 14. If ST has endpoints S(−6, _−1) and T(0, 1), then the midpoint M of ST lies in on the x-axis. -6 + 0 -1 + 1 = M(-3, 0) M , 2 2
-1 + 3 6 + (-2) M( _, _) = M(1, 2)
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―――――――――― √(-1 - (-4)) + (1 - (-3)) = 5
_ _ So, BC ≠ EF. Therefore, BC and EF do not have the same length.
have the same length.
Module 16
―
―――2
2 BC = (2 - 1) + (-1 - (- 4)) = √26
2
© Houghton Mifflin Harcourt Publishing Company •
EF = GH = 5, so EF and GH have the same length. _ _ _ _ 11. AB and CD 12. BC and EF
Concepts and Skills
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another student provide step-by-step instructions for how to copy the line segment. Repeat for finding the midpoint using paper folding. Then have students change roles.
Segment Length and Midpoints
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Show that each statement is true. _ _ 15. If JK has endpoints J(−2, 3) and K(6, 5)_ , and LN _ has endpoints L(0, 7) and N(4, 1), then JK and LN have the same midpoint.
CONNECT VOCABULARY Provide each small group with a deck of note cards with a highlighted term on each, such as collinear, coplanar, line, segment, midpoint, coordinate plane. Provide cards with pictures to illustrate each term. Have students shuffle and deal the cards. Students take turns putting down a card. If the picture or term matches a card another student is holding, that student picks it up and makes a pair. Students must agree on the pairings before play can resume.
_ 16. If GH has endpoints G(-8, 1) and _ H(4, 5), then the midpoint M of GH lies on the line y = −x + 1.
3+ 5 -2 + 6 _ = M (2, 4) , (_ 2 2 ) 0+4 7+ 1 M ( _, _) = M (2, 4) 2 2
M_ JK
_ JK
_ LN
M
1+ 5 -8 + 4 _ = M(-2, 3) , (_ 2 2 )
The midpoint lies on y = −x + 1 since its coordinates satisfy the equation: 3 = −(−2) + 1.
_ LN
Both midpoints have the same coordinates, so the segments have the same midpoint. Use the figure for Exercises 17 and 18.
D
E
AVOID COMMON ERRORS 17. Name two different rays in the figure.
Remind students of the techniques for accuracy in construction, including using a sharp pencil, lining up the compass tip and the pencil tip, and making sure the compass opening stays the same size.
F
18. Name three different segments in the figure.
→ → → ‾ ) and EF ‾ (or DF ‾ DE
_ _ _ _ _ ¯ DE (or ED), EF (or FE), and DF (or FD)
Sketch each figure. 19. two rays that form a straight line and that intersect at point P
20. two line segments that both have a midpoint at point M
© Houghton Mifflin Harcourt Publishing Company
P
M
_ 21. Draw and label a line segment, JK, that is 3 inches long. Use a ruler to draw and label the midpoint M of the segment. J
M
K
22. Draw the segment PQ with endpoints P(−2, −1) and Q(_ 2, 4) on the coordinate plane. Then find the length and midpoint of PQ.
――――――――― 2 2
√(2 - (-2)) + (4 - (-1)) -2 + 2 -1 + 4 M( _, _) = M(0, 1.5)
PQ =
2
4
y
Q
2
― = √41
x -4 -2 P
2
0
2
4
-2 -4
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Lesson 1
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
21–22
1 Recall of Information
MP.4 Modeling
23–25
2 Skills/Concepts
MP.4 Modeling
26
3 Strategic Thinking
MP.3 Logic
27
2 Skills/Concepts
MP.2 Reasoning
28
3 Strategic Thinking
MP.5 Using Tools
29
3 Strategic Thinking
MP.2 Reasoning
30
3 Strategic Thinking
MP.6 Precision
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23. Multi-Step The sign shows distances from a rest stop to the exits for different towns along a straight section of highway. The state department of transportation is planning to build a new exit to Freestone at the midpoint of the exits for Roseville and Edgewood. When the new exit is built, what will be the distance from the exit for Midtown to the exit for Freestone?
Midtown
17 mi
Roseville
35 mi
Edgewood
59 mi
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 When applying the Distance and Midpoint
The distance from the Roseville to Edgewood exits is 59 − 35 = 24 mi, so the distance from the Roseville to 1 24 = 12 mi. The distance from the Freestone exits will be _ 2 Midtown to Roseville exits is 35 − 17 = 18 mi, so the distance from the Midtown to Freestone exits will be 18 + 12 = 30 mi.
Formulas, students may benefit from using different colors to represent the coordinates and the operation symbols. This may help them distinguish the operation from the sign of the coordinate.
24. On a town map, each unit of the coordinate plane represents 1 mile. Three branches of a bank are located at A(−3, 1), B(2, 3), and C(4, −1). A bank employee drives from Branch A to Branch B and then drives halfway to Branch C before getting stuck in traffic. What is the minimum total distance the employee may have driven before getting stuck in traffic? Round to the nearest tenth of a mile.
COGNITIVE STRATEGIES
The minimum total distance occurs when the employee drives along a ― straight line from A to B and from B to the midpoint of BC.
Connect the Midpoint Formula to finding the average of two numbers. In the Midpoint Formula, the midpoint is the mean of the x-coordinates and the y-coordinates.
―
The midpoint N of BC is N(3, 1). ― ― ― ― AB = √29 , BN = √5 , AB + BN = √29 + √5 ≈ 7.6. The minimum total distance the employee may have driven is 7.6 miles. 25. A city planner designs a park that is a quadrilateral with vertices at J(−3, 1), K(1, 3), L(5, −1), and M(−1, −3). There is an entrance to the park at the midpoint of each side of the park. A straight path connects each entrance to the entrance on the opposite side. Assuming each unit of the coordinate plane represents 10 meters, what is the total length of the paths to the nearest meter?
― ― midpoint Q of KL is Q
Midpoint P of JK is P(-1, 2),
(3, 1), _ midpoint R of LM is R(2, -2)
―
PEERTOPEER DISCUSSION
―
―
© Houghton Mifflin Harcourt Publishing Company
midpoint S of MJ is S(-2, -1). ― The total length is √29 + 5 ≈ 10.39, which represents 103.9 meters.
The total length of the paths is approximately 104 meters. 26. Communicate Mathematical Ideas A video game designer places an anthill at the origin of a coordinate plane. A red ant leaves the anthill and moves along a straight line to (1, 1), while a black ant leaves the anthill and moves along a straight line to (−1, −1). Next, the red ant moves to (2, 2), while the black ant moves to (−2, −2). Then the red ant moves to (3, 3), while the black ant moves to (−3, −3), and so on. Explain why the red ant and the black ant are always the same distance from the anthill.
At any given moment, the red ant’s coordinates may be written as (a, a) where a > 0. ――――――― ―― ― 2 2 The red ant’s distance from the anthill is √(a - 0) + (a - 0) = √2a 2 = a √2 . The black ant’s coordinates may be written as (−a, −a) and the black ant’s distance ―――――――― ―― 2 2 from the anthill is √(-a - 0) + (-a - 0) = √2a 2 = a √2 . This shows both ants are always a√2 units from the anthill.
―
―
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Ask students to discuss with a partner how to find the other endpoint of a line segment given one endpoint and the midpoint of the segment. Set the coordinates of the midpoint equal to the total of the missing endpoint and known endpoint coordinates divided by 2. Solve for the missing endpoint coordinates.
The paths are PR and SQ. ― PR = √25 = 5 ― SQ = √29
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Segment Length and Midpoints
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27. Which of the following points are more than 5 units from the point P(-2, -2)? Select all that apply.
JOURNAL Have students compare the Distance and Midpoint Formulas. Ask them to draw an example of each on a grid.
A. A(1, 2)
AP = 5, so AP is not greater than 5.
B. A(3, −1)
BP ≈ 5.1, so BP is greater than 5.
C. A(2, −4)
CP ≈ 4.5, so CP is not greater than 5.
D. A(−6, −6) DP ≈ 5.7, so DP is greater than 5. E. A(−5, 1)
EP ≈ 4.2, so EP is not greater than 5. Answer: B, D
H.O.T. Focus on Higher Order Thinking
28. Analyze Relationships Use a compass and straightedge to construct a segment whose length is AB - CD. Use a ruler to check your construction. B
A C
D
X
Z
Y
―
XZ is the required segment. _ 29. Critical Thinking Point M is the midpoint of AB. The coordinates of point A are (−8, 3) and the coordinates of M are (−2, 1). What are the coordinates of point B?
Let (x, y) be the coordinates of point B. -8 + x Solve for x: -2 = , 4= x 2 The coordinates of point B are (4, −1).
© Houghton Mifflin Harcourt Publishing Company
_
Conjecture Use a compass and straightedge 30. Make a_ to copy AB so that one endpoint of the copy is at point X. Then repeat the process _three more times, making three different copies of AB that have an endpoint at point X. Make a conjecture_ about the set of all possible copies of AB that have an endpoint at point X. A
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Lesson 16.1
1=
3 +y _ , -1 = y 2
X
B
Constructions may vary. Possible answer: ― The set of all possible copies of AB that have an endpoint at point X form a circle and its interior. The center of the circle is X and the radius is AB.
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Solve for y:
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Students can check that they have found the
A carnival ride consists of four circular cars—A, B, C, and D—each of which spins about a point at its center. The center points of cars A and B are attached by a straight beam, as are the center points of cars C and D. The two beams are attached at their midpoints by a rotating arm. The figure shows how the beams and arm can rotate.
correct coordinates of P by seeing if the x-coordinate of P is midway between the x-coordinates of points A and B, and if the y-coordinate of P is midway between the y-coordinates of points A and B:
C A
D
3m
B
A(-6, -1)
P(-4, -2)
A plan for the ride uses a coordinate plane in which each unit represents one meter. In the plan, the center of car A is (−6, −1), the center of car B is (−2, −3), the center of car C is (3, 4), and the center of car D is (5, 0). Each car has a diameter of 3 meters.
B(-2, -3)
They can use the same method to check the coordinates of Q:
The manager of the carnival wants to place a fence around the ride. Describe the shape and dimensions of a fence that will be appropriate to enclose the ride. Justify your answer.
―
―
C(3, 4)
Let P be the midpoint of AB and let Q be the midpoint of CD. The coordinates of P and Q are P(-4, -2). and Q(4, 2). ― Find the length of the rotating arm PQ. PQ = √80
Q(4, 2) D(5, 0)
The maximum length of the ride occurs when the two beams lie along the rotating arm, as shown. A
P
B
C
Q
D
1.5 m
AVOID COMMON ERRORS
© Houghton Mifflin Harcourt Publishing Company
1.5 m
This process helps to make the calculation of midpoints more logical and less prone to error.
― ― AP = √5 and BP = √5 since AP = BP. ― ― CQ = √5 and DQ = √5 since CQ = DQ. The total length L is AP + PQ + QD plus half the diameter of a car on either end. ― ― ― L = 1.5 + √5 + √80 + √5 + 1.5 ≈ 16.4 m. Students’ descriptions should allow extra space for clearance around the ride. Possibilities include a square fence that is about 18 meters long on each side or a circular fence about 18 meters in diameter.
When finding PQ using the Distance Formula, students may calculate
――― 64 + 16 incorrectly: ――― 64 + 16 = ― 64 + ― 16
=8+4 = 12
Stress that by the order of operations, sums and differences beneath a square root sign must be calculated first, before the square root is taken:
――― 64 + 16 = ――― 64 + 16 80 = ―
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Have students graph two versions of the carnival ride on coordinate grids, using the given coordinates and scale. Ask them to show the ride in these configurations:
4/19/14 10:16 AM
• when it is at its maximum horizontal length • when it is at its maximum vertical height When students have completed the two graphs, they should draw the fence that they described in the Lesson Performance Task and calculate its length.
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Segment Length and Midpoints
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LESSON
16.2
Name
Angle Measures and Angle Bisectors
Class
Date
16.2 Angle Measures and Angle Bisectors Essential Question: How is measuring an angle similar to and different from measuring a line segment?
Common Core Math Standards
Resource Locker
The student is expected to: COMMON CORE
G-CO.A.1
Explore
Know precise definitions of angle ... based on the undefined notions of … distance around a circular arc. Also G-CO.D.12
Constructing a Copy of an Angle
Start with a point X and use a compass and straightedge to construct a copy of ∠S. Z
Mathematical Practices COMMON CORE
S
MP.5 Using Tools
Language Objective
Use a straightedge to draw a ray with endpoint X.
Work with a partner to play “angle charades.”
Place the point of your compass on S and draw an arc that intersects both sides of the angle. Label the points of intersection T and U.
ENGAGE
S
© Houghton Mifflin Harcourt Publishing Company
Possible answer: In both cases, the measure is undefined until a unit is chosen. Angles may be measured in degrees; there are 360° in a circle. The tool for measuring an angle in degrees is a protractor. Line segments are measured using linear units, such as centimeters or inches. The tool for measuring a line segment is a ruler.
Place the point of the compass on T and open it to the distance TU.
U
U
Essential Question: How is measuring an angle similar to and different from measuring a line segment?
Y
X
S
T
Without adjusting the compass, place the point of the compass on X and draw an arc that intersects the ray. Label the intersection Y.
T
Without adjusting the compass, place the point of the compass on Y and draw an arc. Label the intersection with the first arc Z. → ‾ . Use a straightedge to draw XZ ∠X is a copy of ∠S.
Reflect
1.
→ → ‾ , ‾ coincides with ST If you could place the angle you drew on top of ∠S so that XY → ‾ ? Explain. what would be true about XZ → → ‾ . Since the angles are copies of each other, the rays in each ‾ would coincide with SU XZ
angle form the same opening.
PREVIEW: LESSON PERFORMANCE TASK
2.
View the online Engage. Discuss the photo and the fact that 60° and 40° stands are available, but that a customer wants a 50° stand. Then preview the Lesson Performance Task.
Module 16
Discussion Is it possible to do the construction with a compass that is stuck open to a fixed distance? Why or why not? No; you could use the compass to make the required arcs in Steps B and C, but you would not be able to adjust the opening of the compass as required in Step D. be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Lesson 2
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Date Class
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COMMON CORE
IN1_MNLESE389762_U7M16L2 789
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→ ‾ , → coincides with ST ‾ XY ∠S so that rays in each on top of other, the you drew angle→ s of each place the ? Explain. s are copie ‾ → If you could true about XZ the angle be ‾ . Since with SU what would → would coincide ‾ XZ ing. stuck open same open ss that is the compa a would with angle form but you construction B and C, to do the arcs in Steps it possible Is red not? ssion the requi Why or why 2. Discu Lesson 2 in Step D. ass to make distance? as required to a fixed the comp compass could use No; you ing of the the open 789 to adjust not be able
n Mifflin © Houghto
Module 16
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62_U7M1
Lesson 16.2
point ss, place the the the compa Label adjusting draw an arc. Without ss on Y and arc Z. of the compa with the first intersection → ‾ . XZ tedge to draw Use a straigh of ∠S. ∠X is a copy
Reflect
1.
ESE3897
IN1_MNL
open it
point Place the ce TU. to the distan
T
the point
Watch for the hardcover student edition page numbers for this lesson.
Y
ss on T and of the compa
U
S
HARDCOVER PAGES 789800
copy of ∠S.
X S
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Resource Locker
Quest Essential
4/19/14
10:35 AM
4/19/14 10:34 AM
Naming Angles and Parts of an Angle
Explain 1
EXPLORE
An angle is a figure formed by two rays with the same endpoint. The common endpoint is the vertex of the angle. The rays are the sides of the angle. Example 1
Constructing a Copy of an Angle
Draw or name the given angle.
∠PQR
P
When an angle is named with three letters, the middle letter is the vertex. So, the vertex of angle ∠PQR is point Q.
QUESTIONING STRATEGIES
Q
Do the rays of the angle you construct need to be the same length as the rays of the given angle? Why or why not? No; the measure of the angle is determined only by the size of the opening between the rays, not by the lengths of the rays.
R
The sides of the angle are two rays with common endpoint Q. So, → → ‾ and QR ‾ . the sides of the angle are QP Draw and label the angle as shown.
J
When you draw the initial arc that intersects the side of the angle to be copied, does it matter how wide you open the compass? Explain. No, as long as the arc intersects both sides of the angle, it doesn’t matter.
1 L
K
The vertex of the angle shown is point K . A name for the angle is ∠ K . The vertex must be in the middle, so two more names for the angle are ∠ J and ∠ L
K
L
K
J .
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Have students practice constructing both
The angle is numbered, so another name is ∠ 1 . Reflect
3.
© Houghton Mifflin Harcourt Publishing Company
Without seeing a figure, is it possible to give another name for ∠MKG? If so, what is it? If not, why not? Yes; ∠GKM
Your Turn
B
Use the figure for 4–5. 4.
Name ∠2 in as many different ways as possible.
2
∠AEB, ∠BEA 5.
C
A
3 E
4 D
Use a compass and straightedge to copy ∠BEC.
acute and obtuse angles.
EXPLAIN 1 Naming Angles and Parts of an Angle CONNECT VOCABULARY
Module 16
790
Lesson 2
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M16L2 790
Math Background
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Connect the word degree to the idea of measurement. A degree in science may be a measure of temperature in units known as Fahrenheit or Celsius. Degree in this context is the measure of an angle. Ask how many degrees are in a straight angle, a right angle, and so on.
Compass and straightedge constructions date to ancient Greece. In fact, one of the classic problems of ancient Greek mathematics was the trisection of an angle. That is, using a compass and straightedge, is it possible to construct an angle whose measure is one-third that of an arbitrary given angle? It was not until 1837 that this construction was proven to be impossible. On the other hand, it is a straightforward task to bisect any angle, and students learn this fundamental construction in this lesson.
Angle Measures and Angle Bisectors 790
Explain 2
QUESTIONING STRATEGIES
Measuring Angles
The distance around a circular arc is undefined until a measurement unit is chosen. Degrees (°) are a common 1 measurement unit for circular arcs. There are 360° in a circle, so an angle that measures 1° is ___ of a circle. 360 The measure of an angle is written m∠A or m∠PQR.
When an angle is named using three letters, how can you identify the vertex of the angle? The vertex is the center letter of the angle name.
You can classify angles by their measures.
Classifying Angles Acute Angle
An angle diagram may use letters or numbers to identify the angle. How are the diagrams different? Letters label individual points on the angle, while a number is inside the angle and names the entire angle.
0° < m∠A < 90°
EXPLAIN 2
Measuring Angles
Obtuse Angle
m∠A = 90°
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Suggest that students use a straightedge, such as an index card, to extend the rays of an angle before they use a protractor to measure the angle. If the angle is smaller than the distance from the center mark to the edge of the protractor, this will make it easier to accurately measure the angle. Encourage students to estimate an angle measure before measuring to make sure the measurement is reasonable.
m∠A = 180°
Use a protractor to draw an angle with the given measure.
53°
→ ‾ . Step 1 Use a straightedge to draw a ray, XY Y
Step 2 Place your protractor on point X as shown. Locate the point along the edge of the protractor that corresponds to 53°. Make a mark at this location and label it point Z. Z
© Houghton Mifflin Harcourt Publishing Company
Remind students to place the center mark of the protractor on the vertex and to align one side of the angle with the 0° mark. They may have to rotate the angle or the protractor for ease of alignment. On some protractors, the zero line is on the bottom edge, while on others, it is placed higher.
A
90° < m∠A < 180°
X
AVOID COMMON ERRORS
Straight Angle
A
A
A
Example 2
Right Angle
X Y
→ ‾ . m∠ZXY = 53°. Step 3 Draw XZ Z
X
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Small Group Activity Use pictures from magazines to find angles of different sizes. Ask students to identify the type of angle and estimate the measure. Then have students measure the angles with a protractor. If protractors are not available, they can use index cards or origami paper. The edges are already at a 90° angle, and anything greater would be an obtuse angle. A half-fold forms a 45° angle, a tri-fold approximately 30°, and so on. The pictures can be posted by classification and used for reference.
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B
138°
QUESTIONING STRATEGIES C A
If the vertex of an angle is placed on the center point of a protractor and both rays of the angle lie within the measures of the protractor, does one of the rays have to align with the 0° mark to find the measure of the angle? Explain. No, you can find the absolute value of the difference of the measures each ray intersects to find the measure of the angle. For example, if one ray aligns with 25° and the other with 67°, the angle measures 42°.
B
→ ‾ . Step 1 Use a straightedge to draw a ray, AB
→ ‾ is at zero. Step 2 Place your protractor on point A so that AB Step 3 Locate the point along the edge of the protractor that corresponds to 138°. Make a mark at this location and label it point C. → ‾ . m∠CAB = 138°. Step 4 Draw AC
Reflect
6.
Explain how you can use a protractor to check that the angle you constructed in the Explore is a copy of the given angle. Measure the given angle and the constructed angle. They should have the same measure.
EXPLAIN 3
Your Turn
Constructing an Angle Bisector
Each angle can be found in the rigid frame of the bicycle. Use a protractor to find each measure. 7.
J
8.
M
CONNECT VOCABULARY © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Gena73/ Shutterstock
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L
40°
Explain 3
105°
Constructing an Angle Bisector
An angle bisector is a ray that divides an angle into two angles that both have the → ‾ bisects ∠ABC, so m∠ABD = m∠CBD. The arcs same measure. In the figure, BD in the figure show equal angle measures.
A D
Postulate 2: Angle Addition Postulate If S is in the interior of ∠PQR, then m∠PQR = m∠PQS + m∠SQR.
B
S
P
C
The postulates for angles are similar to the postulates for segments. The Protractor Postulate is similar to the Ruler Postulate. It says that the measure of an angle is the absolute value of the difference between the numbers matched on a protractor with the rays that form the sides of the angle.
R Q
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Manipulatives
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Have students investigate how to find the bisector of an angle using a geometric reflecting tool. Have students draw an angle on a piece of paper. To use the tool, place it on the vertex of the angle so that one side is reflected onto the other side. Then draw the tool’s line. Discuss how using the reflective device is similar to using paper folding to find the angle bisector.
Angle Measures and Angle Bisectors 792
Example 3
AVOID COMMON ERRORS Remind students not to change the compass setting when they draw the intersecting arcs from each side ray of an angle to create the angle bisector. In order to help students see why this is important, you many want to have them do a construction in which they change the compass setting between arcs. Students will see that the resulting ray does not bisect the angle.
Use a compass and straightedge to construct the bisector of the given angle. Check that the measure of each of the new angles is one-half the measure of the given angle.
M
Step 1 Place the point of your compass on point M. Step 2 Place the point of the compass on P and Draw an arc that intersects both sides of the draw an arc in the interior of the angle. angle. Label the points of intersection P and Q. P
P
M
QUESTIONING STRATEGIES If a ray divides an angle into two angles with equal measures, what must be true about the ray? Explain. The ray is the angle bisector of the angle by the definition of an angle bisector.
M
Q
Q
→ ‾ . Step 4 Use a straightedge to draw MR
Step 3 Without adjusting the compass, place the point of the compass on Q and draw an arc that intersects the last arc you drew. Label the intersection of the arcs R.
P
P
R
R
VISUAL CUES
M
© Houghton Mifflin Harcourt Publishing Company
Some students may have difficulty visualizing two angles that have the same measure, especially if the sides of the angles are shown with rays of different lengths. You may want to have students construct angle copies on tracing paper. Then they can place the copy on top of the original angle to check that the measures are the same.
M
Q
Q
Step 5 Measure with a protractor to confirm that m∠PMR = m∠QMR = _12m∠PMQ. 1 (54°)✓ 27° = 27° = _ 2 A
B
D C
Step 1 Draw an arc centered at A that intersects both sides of the angle. Label the points of intersection B and C. Step 2 Draw an arc centered at B in the interior of the angle. Step 3 Without adjusting the compass, draw an arc centered at C that intersects the last arc you drew. Label the intersection of the arcs D. → ‾ . Step 4 Draw AD
_1 Step 5 Check that m∠BAD = m∠CAD = _12m∠BAC. Yes; 45° = 45° = 2 (90°)
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Connect Vocabulary Remind students that the prefix bi- means “two” and that the root sect means “to cut.” They can use these cues to help them remember that an angle bisector divides the angle into two equal parts.
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Reflect
9.
ELABORATE
Discussion Explain how you could use paper folding to construct the bisector of an angle. Fold the paper so that one side of the angle lies on top of the other. Unfold the paper.
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students to record angle measures
The crease is the angle bisector. Your Turn
Use a compass and straightedge to construct the bisector of the given angle. Check that the measure of each of the new angles is one-half the measure of the given angle. 10.
using a protractor in degrees by using the degree symbol. Point out that not all angle measures are recorded in degrees. Radians are real number units of angle rotation. For example, π radians = 180°.
11.
Elaborate
INTEGRATE TECHNOLOGY
12. What is the relationship between a segment bisector and an angle bisector? A segment bisector divides a line segment into two segments that have the same length;
Point out that a graphing calculator may need to be set to record angle measure in degrees, since either degree or radian measure can be selected. This feature is generally used for trigonometry calculations, however.
an angle bisector divides an angle into two angles that have the same measure.
13. When you copy an angle, do the lengths of the segments you draw to represent the two rays affect whether the angles have the same measure? Explain. No; the measure of an angle depends only on the portion of a circle that the angle
encompasses, not upon the apparent length of its sides.
is acute, use the measure between 0° and 90°. If the angle is obtuse, use the measure between 90° and 180°.
© Houghton Mifflin Harcourt Publishing Company
14. Essential Question Check-In Many protractors have two sets of degree measures around the edge. When you measure an angle, how do you know which of the two measures to use? Answers may vary. Sample: First determine if the angle is acute or obtuse. If the angle
QUESTIONING STRATEGIES What methods can you use to bisect an angle? Which method do you think is the most accurate? Explain. You can use a compass and straightedge, paper folding, or measurement with a protractor. Possible answer: You are more likely to draw the bisector accurately from the vertex by using a compass and straightedge because the method is exact.
SUMMARIZE THE LESSON Module 16
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What is the Angle Addition Postulate and how does it relate to the bisector of an angle? If a ray from the vertex of an angle divides the angle into two parts, the sum of the measures of the parts is equal to the measure of the whole original angle. An angle bisector is a ray that divides an angle into two equal parts.
Angle Measures and Angle Bisectors 794
Evaluate: Homework and Practice
EVALUATE
• Online Homework • Hints and Help • Extra Practice
Use a compass and straightedge to construct a copy of each angle. 1.
2.
3.
ASSIGNMENT GUIDE Concepts and Skills
Practice
Explore Constructing a Copy of an Angle
Exercises 1–3
Example 1 Naming Angles and Parts of an Angle
Exercises 4–7
Example 2 Measuring Angles
Exercises 8–11
Example 3 Constructing an Angle Bisector
Draw an angle with the given name. ∠JWT
4.
5.
Exercises 12–14 W
applied in addition to the Angle Addition Postulate to find the measure of angles outlined on top of a protractor.
KINESTHETIC EXPERIENCE Have students work in pairs to write highlighted and prerequisite vocabulary from the lesson on index cards, such as acute, obtuse, and straight angles; angle bisector, and ray. Students talk about what each term means, then place the cards face down. One student draws a card and “acts out” the term on the card (for example, a right angle) using hands and arms. The other student guesses. Then they switch roles and the first student guesses while the second student acts out or draws the term.
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Name each angle in as many different ways as possible. 6. W © Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss how the Protractor Postulate can be
∠NBQ
J
7.
X
G
1
2 L
J
Z
∠W, ∠ZWX, ∠XWZ, and ∠1
∠L, ∠GLJ, ∠JLG, and ∠2
Use a protractor to draw an angle with the given measure. 8.
19°
9. C
A
100° C
B A
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–3
1 Recall of Information
MP.5 Using Tools
4–7
1 Recall of Information
MP.6 Precision
8–14
1 Recall of Information
MP.5 Using Tools
15–16
2 Skills/Concepts
MP.2 Reasoning
17–19
2 Skills/Concepts
MP.5 Using Tools
20–21
2 Skills/Concepts
MP.4 Modeling
22
2 Skills/Concepts
MP.4 Modeling
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Use a protractor to find the measure of each angle. 10.
VISUAL CUES
11.
Remind students to show all arcs and extend segments far enough when creating compass and straightedge constructions.
P
Q
172°
55°
F
E
D R
AVOID COMMON ERRORS
Use a compass and straightedge to construct the bisector of the given angle. Check that the measure of each of the new angles is one-half the measure of the given angle. 12.
13.
If students’ compass settings are not tightly fixed, the compass setting may change without students’ awareness. Stress to students that they must check compass tightness and keep the same fixed compass setting for accuracy.
14.
Use the Angle Addition Postulate to find the measure of each angle. 15. ∠BXC
C
m∠AXB + m∠BXC = m∠AXC
D
B
40° + m∠BXC = 70°
E
m∠BXC = 30° X
A
16. ∠BXE
F
m∠EXF + m∠BXE = m∠BXF © Houghton Mifflin Harcourt Publishing Company
30° + m∠BXE = 140° m∠BXE = 110°
Use a compass and straightedge to copy each angle onto a separate piece of paper. Then use paper folding to construct the angle bisector. 17.
18.
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
23
2 Skills/Concepts
MP.4 Modeling
24
2 Skills/Concepts
MP.6 Precision
25
2 Skills/Concepts
MP.2 Reasoning
26
3 Strategic Thinking
MP.2 Reasoning
27
3 Strategic Thinking
MP.3 Logic
28
2 Skills/Concepts
MP.5 Using Tools
29
3 Strategic Thinking
MP.5 Using Tools
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Angle Measures and Angle Bisectors 796
19. Use a compass and straightedge to construct an angle whose measure is m∠A + m∠B. Use a protractor to check your construction.
COMMUNICATING MATH Students will have to use algebra together with the Angle Addition Postulate to set up an equation to solve for the variable when one of the angles includes a variable. Point out that angles may not always have whole-number measures.
B
A
20. Find the value of x, given that m∠PQS = 112°. R
21. Find the value of y, given that m∠KLM = 135°.
47° (10x)°
72° P
M
L
S
(16y)°
K N
Q
m∠KLN + m∠NLM = m∠KLM
m∠PQR + m∠RQS = m∠PQS
47 + 16y = 135
72 + 10x = 112
y = 5.5
x=4
© Houghton Mifflin Harcourt Publishing Company
22. Multi-Step The figure shows a map of five streets that meet at Concord Circle. The measure of the angle formed by Melville Road and Emerson Avenue is 118°. The measure of the angle formed by Emerson Avenue and Thoreau Street is 134°. Hawthorne Lane bisects the angle formed by Melville Road and Emerson Avenue. Dickinson Drive bisects the angle formed by Emerson Avenue and Thoreau Street. What is the measure of the angle formed by Hawthorne Lane and Dickinson Drive? Explain your reasoning. Hawthorne Ln.
Melville Rd. Concord Circle
The measure of the angle formed by Melville and Emerson is 118°, so the 1( measure of the angle formed by Hawthorne and Emerson is __ 118°) = 59°. 2 The measure of the angle formed by Emerson and Thoreau is 134°, so the 1( 134°) = 67°. By measure of the angle formed by Emerson and Dickinson is __ 2 the Angle Addition Postulate, the measure of the angle formed by Hawthorne and Dickinson is 59° + 67° = 126°.
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Thoreau St.
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23. Represent Real-World Problems A carpenter is building a rectangular bookcase with diagonal braces across the back, as shown. The carpenter knows that ∠ADC is a right angle and that m∠BDC is 32° greater than m∠ADB. Write and solve an equation to find m∠BDC and m∠ADB.
A
B
m∠ADB + m∠BDC = m∠ADC
CRITICAL THINKING Review why students can use the Angle Addition Postulate to find a missing angle measure if they know the measures of one angle and the total angle, in order to find the measure of the other angle, when an angle is divided into two angles that do not overlap.
E
x + (x + 32) = 90 2x + 32 = 90
x = 29
So, m∠ADB = 29° and m∠BDC = 29 + 32 = 61°
D
C
MODELING 24. Describe the relationships among the four terms.
Students know the measure of a right angle. Discuss how to use the measure of a right angle to find the measure of a straight angle (180°) and the total number of degrees in one full rotation (360°).
Angle bisector
The definitions of the terms “angle bisector” and “angle” are each built upon the definitions of the term below it. The definition of the term “ray” is built upon the undefined term “line” below it.
Defined terms
Angle Ray
25. Determine whether each of the following pairs of angles have equal measures. Select the correct answer for each lettered part. A. ∠KJL and ∠LJM B. ∠MJP and ∠PJR C. ∠LJP and ∠NJR
E. ∠KJR and ∠MJP
No
Yes
No
Yes
No
Yes
No
Yes
No
Have students work with a partner to write a guide to copying and bisecting angles in the form of a comic strip. Encourage them to include enough information so that someone who has never done these constructions could follow the procedure.
M N
L 46° 42°
P
J 48° 46° Q
K © Houghton Mifflin Harcourt Publishing Company
D. ∠MJK and ∠PJR
Yes
PEERTOPEER DISCUSSION
Undefined term
Line
R
a. no; m∠LJM = 90° - 42° = 48° ≠ m∠KJL b. yes; m∠NJP = 48° so m∠MJP = 46° + 48° = 94° and m∠PJR = 48° + 46° = 94° c. yes; m∠NJP = 48° and m∠LJM = 90° - 42° = 48°, so m∠LJP = 48° + 46° + 48° = 142° and m∠NJR = 48° + 48° + 46° = 142° d. no; m∠MJK = 90°, but m∠PJR = 48° + 46° = 94° e. no; m∠KJR = 360° - 90° - 46° - 48° - 48° - 46° = 82°, but m∠MJP = 46° + 48° = 94°
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26. Make a Conjecture A rhombus is a quadrilateral with four sides of equal length. Use a compass and straightedge to bisect one of the angles in each of the rhombuses shown. Then use your results to state a conjecture.
JOURNAL Have students name and give the definition of at least five other words that use the prefix bi- to mean “two.”
Constructions may vary. Sample:
In a rhombus, the bisector of an angle also bisects the opposite angle. H.O.T. Focus on Higher Order Thinking
27. What If? What happens if you perform the steps for constructing an angle bisector when the given angle is a straight angle? Does the construction still work? If so, explain why and show a sample construction. If not, explain why not.
Yes; the construction still works. In this case, the construction produces two right angles since each has half the measure of a straight angle (180°).
© Houghton Mifflin Harcourt Publishing Company
28. Critical Thinking Use a compass and straightedge to construct an angle whose measure is m∠A - m∠B. Use a protractor to check your construction.
A
29. Communicate Mathematical Ideas Explain the steps for using a compass and straightedge to construct an angle with __14 the measure of a given angle. Then draw an angle and show the construction.
Construct the bisector of the given angle. Then construct the bisector of one of the angles that was formed.
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Lesson Performance Task
QUESTIONING STRATEGIES You are given a 60° angle and an 80° angle. How could you use them to construct a 20° angle? Sample answer: Copy the 60° angle inside the 80° angle, with the two angles sharing a side. The angle adjacent to the 60° angle will measure 80° – 60° = 20°. You are given a 70° angle and a 60° angle. How could you use them to construct a 25° angle? Sample answer: Bisect the 70° angle to create two 35° angles. Copy a 35° angle inside the 60° angle, with the two angles sharing a side. The angle adjacent to the 35° angles will measure 60° – 35° = 25°.
A store sells custom-made stands for tablet computers. When an order comes in, the customer specifies the angle at which the stand should hold the tablet. Then an employee bends a piece of aluminum to the correct angle to make the stand. The figure shows the templates that the employee uses to make a 60° stand and a 40° stand.
60°
You are given a 50° angle and a 40° angle. How could you use them to construct a 5° angle? Sample answer: Copy the 40° angle inside the 50° angle, with the two angles sharing a side. The angle adjacent to the 40° angle will measure 50° – 40° = 10°. Then bisect the 10° angle to create two 5° angles.
40°
The store receives an order for a 50° stand. The employee does not have a template for a 50° stand and does not have a protractor. Can the employee use the existing templates and a compass and straightedge to make a template for a 50° stand? If so, explain how and show the steps the employee should use. If not, explain why not.
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© Houghton Mifflin Harcourt Publishing Company
Yes; first construct the bisector of the template for the 60° stand to create two 30° angles. Then construct the bisector of the template for the 40° stand to create two 20° angles. Next, copy one of the 30° angles. Finally, copy one of the 20° angles so it shares a side with the 30° angle. The measure of the resulting angle is 20° + 30° = 50°.
AVOID COMMON ERRORS Students may have difficulty completing the last step of the Lesson Performance Task, in which they must copy a 20° angle so that it shares the non-horizontal side of the 30° angle. This can happen when students are used to starting with horizontal lines in their constructions. Point out that there is nothing wrong with rotating their papers to start with a horizontal line.
Lesson 2
EXTENSION ACTIVITY IN1_MNLESE389762_U7M16L2 800
The Lesson Performance Task introduces the idea of combining simple constructions to produce more complex ones. Have students think about how they could use this idea to construct a 35° angle from a 30° angle and a 40° angle. Then have them make up problems that apply the idea. Each problem should give the measures of two or more angles and ask how an angle of specified measure could be constructed using the given ones. Students should provide answers for each of their problems.
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Angle Measures and Angle Bisectors 800
LESSON
16.3
Name
Representing and Describing Transformations
Class
16.3 Representing and Describing Transformations Essential Question: How can you describe transformations in the coordinate plane using algebraic representations and using words?
Common Core Math Standards G-CO.A.2
A transformation is a function that changes the position, shape, and/or size of a figure. The inputs of the function are points in the plane; the outputs are other points in the plane. A figure that is used as the input of a transformation is the preimage. The output is the image. Translations, reflections, and rotations are three types of transformations. The decorative tiles shown illustrate all three types of transformations.
Represent transformations in the plane ...; describe transformations as functions … . Compare transformations that preserve distance and angle to those that do not … . Also G-CO.B.5
You can use prime notation to name the image of a point. In the diagram, the transformation T moves point A to point A′ (read “A prime”). You can use function notation to write T( A ) = A′. Note that a transformation is sometimes called a mapping. Transformation T maps A to A′.
Mathematical Practices COMMON CORE
MP.6 Precision
Language Objective
A'
Students work together to give oral, verbal and pictorial clues and justify transformations drawn from clues.
T
Essential Question: How can you describe transformations in the coordinate plane using algebraic representations and using words? Possible answer: You can use coordinate notation to write rules that describe how preimage points are transformed to image points. You can describe transformations with words like reflection, rotation, translation, stretch, and dilation. You can say that a transformation is rigid (preserves length and angle measure) or not rigid (does not preserve length and angle measure.)
A
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Antony McAulay/Shutterstock
ENGAGE
Resource Locker
Performing Transformations Using Coordinate Notation
Explore
The student is expected to: COMMON CORE
Date
Image
Preimage
Coordinate notation is one way to write a rule for a transformation on a coordinate plane. The notation uses an arrow to show how the transformation changes the coordinates of a general point, ( x, y ).
Find the unknown coordinates for each transformation and draw the image. Then complete the description of the transformation and compare the image to its preimage.
(x, y) → (x - 4, y - 3) Preimage (x, y) A(0, 4)
→
B(3, 0) C( 0, 0 )
Rule (x, y) → (x - 4, y - 3)
5
Image (x - 4, y - 3)
A′(0 − 4, 4 − 3)
=
→
B′(3 − 4, 0 − 3)
=
→
C′( 0 − 4, 0 − 3 )
=
A′( −4, 1 )
y
A A'
⎛ ⎞ B′ ⎜ -1 , -3 ⎟ ⎝ ⎠ ⎛ ⎞ C′ ⎜ -4 , -3 ⎟ ⎝ ⎠
-5 C'
The transformation is a translation 4 units (left/right)
x 0 C
B
5
B' -5
and 3 units (up/down). A comparison of the image to its preimage shows that Possible answer: the image is the same size and shape as the preimage
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Lesson 3
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made throu
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Date Class
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HARDCOVER PAGES 801814
Resource Locker
Quest Essential COMMON CORE
Perform The Notation of a figure. and/or size Coordinate plane. A n, shape,
Explore
the s the positio are other points in is the image. n that change output outputs ion is a functio in the plane; the is the preimage. The The decorative tiles A transformatfunction are points rmation rmations. the of a transfo types of transfo inputs of the input ns are three is used as ns. rmation figure that reflections, and rotatio , transformatio m, the transfo T( A) = A′. Translations te all three types of In the diagra to write of a point. notation to A′. shown illustra function the image use T maps A name can to ion n ). You ormat prime notatio “A prime” ng. Transf You can use A to point A′ (read imes called a mappi point n is somet T moves a transformatio Note that A' Image
Watch for the hardcover student edition page numbers for this lesson.
T A notation Preimage plane. The a coordinate point, ( x, y ). l rmation on for a transfo coordinates of a genera write a rule s the image. Then age. one way to rmation change draw the notation is the transfo to its preim tion and Coordinate to show how transforma compare the image for each uses an arrow
coordinates transformation and y unknown the 5 Find the iption of the descr complete A - 3) Image (x - 4, y 3) (x, y) → (x - 4, y Rule A' - 3) (x - 4, y ) Preimage A′( −4, 1 (x, y) → 0 C = (x, y) ) 3 − 4 ⎞ -5 A′(0 − 4, ⎛ → -3 ⎟ B′ ⎜ -1 , A(0, 4) ⎠ = ⎝ B' 0 − 3) ⎞ B′(3 − 4, C' ⎛ → -3 ⎟ -5 B(3, 0) C′ ⎜ -4 , ⎠ = ⎝ 0 − 3) C′( 0 − 4, → C( 0, 0 ) (left/right) tion 4 units is a transla ormation transf The age that (up/down). the preim age shows and 3 units shape as to its preim and image size rison of the e is the same A compa the imag answer: Possible
Credits:
IN1_MNLESE389762_U7M16L3 801
y • Image g Compan
View the online Engage. Discuss the photo and the fact that it has been stretched. Ask whether the angles in the photo match the real-life angles. Then preview the Lesson Performance Task.
ng
d Describi senting an 16.3 Repre tions Transforma
Name
Publishin Harcour t n Mifflin © Houghto /Shutterstock McAulay ©Antony
PREVIEW: LESSON PERFORMANCE TASK
x B
5
. Lesson 3
801 Module 16
6L3 801
62_U7M1
ESE3897
IN1_MNL
801
Lesson 16.3
4/19/14
12:00 PM
4/19/14 11:58 AM
B
(x, y) → (-x, y) Rule
Preimage (x, y)
Image
(x, y) → (-x, y) R′(-(−4), 3)
=
→
S′(-(−1), 3)
=
→
T′( -(−4), 1 )
=
R(-4, 3)
→
S(-1, 3) T(-4, 1)
5
(-x, y) ⎛ ⎞ R′⎜ 4 , 3 ⎟ ⎝ ⎠ ⎛ ⎞ S′⎜ 1 , 3 ⎟ ⎝ ⎠ ⎛ ⎞ T′⎜ 4 , 1 ⎟ ⎝ ⎠
R
EXPLORE
y
S S'
R'
0
T' x 5
T -5
Performing Transformations Using Coordinate Notation INTEGRATE TECHNOLOGY
-5
The transformation is a reflection across the (x-axis/y-axis).
You can use a spreadsheet to enter coordinates of vertices to create geometric figures and then transform the coordinates using a formula to create transformed figures.
A comparison of the image to its preimage shows that
Possible answer: the image is the same size and shape as the preimage, but it is flipped over the y-axis
C
.
(x, y) → (2x, y) Preimage (x, y)
⎛ ⎞ J ⎜ -1 , 2 ⎟ → ⎝ ⎠ ⎛ ⎞ K⎜ 2 , 2 ⎟ → ⎝ ⎠ ⎛ ⎞ L ⎜ 2 , -4 ⎟ → ⎝ ⎠
Rule
(x, y) → (2x, y) ⎛ ⎞ J ′ ⎜2 ⋅ -1 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ K′ ⎜2 ⋅ 2 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ L′ ⎜2 ⋅ 2 , -4 ⎟ ⎝ ⎠
Image
5
(2x, y) ⎛ ⎞ = J ′ ⎜ -2 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ = K′ ⎜ 4 , 2 ⎟ ⎝ ⎠ ⎛ ⎞ = L′ ⎜ 4 , -4 ⎟ ⎝ ⎠
J' J
K
K' x
-5
The transformation is a (horizontal/vertical) stretch by a factor of 2 .
0
-5
5
L
student is the preimage. The image is the reflection of the student in the mirror. Ask students to use items on their desks to demonstrate a translation and a rotation. For each transformation, have them identify the preimage and the image.
L'
Possible answer: the image and the preimage are both right triangles, but they do not have the same size or shape
.
Reflect
Discussion How are the transformations in Steps A and B different from the transformation in Step C? The transformations in Steps A and B preserve the size and shape of the right triangle. The transformation in Step C changes the shape of the right triangle.
2.
© Houghton Mifflin Harcourt Publishing Company
A comparison of the image to its preimage shows that
1.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students stand in front of a mirror. The
y
QUESTIONING STRATEGIES How is the notation for an image related to the notation for its preimage? It is the same except that the notation for the image has a prime mark after each letter. How is the notation T(A) = Aʹ similar to the more familiar function notation y = ƒ(x)? The object inside the parentheses is the input. The object on the other side of the equal sign is the output.
For each transformation, what rule could you use to map the image back to the preimage? A. (x, y) → (x + 4, y + 3); B. (x, y) → (-x, y); C. (x, y) → (0.5x, y)
Module 16
802
Lesson 3
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M16L3 802
Integrate Mathematical Practices
This lesson provides an opportunity to address Mathematical Practice MP.6, which calls for students to “attend to precision.” Students are already familiar with transformations in the plane and, in this lesson, students use graph paper to draw transformations. They use protractors, rulers, and coordinates to determine whether length and angle measure have been preserved. They also use concepts about functions to write the rules that express transformations algebraically.
4/19/14 11:57 AM
How can you use the rule for transforming the coordinates of a general point of a figure to help you recognize if the transformation changes the size of the image? If a coordinate is changed by a multiplicative factor other than –1, the image will change in size.
CONNECT VOCABULARY Emphasize that the rule for the transformation shows how to change each preimage coordinate to its corresponding image coordinate.
Representing and Describing Transformations
802
EXPLAIN 1
Some transformations preserve length and angle measure, and some do not. A rigid motion (or isometry) is a transformation that changes the position of a figure without changing the size or shape of the figure. Translations, reflections, and rotations are rigid motions.
Describing Rigid Motions Using Coordinate Notation
Properties of Rigid Motions
AVOID COMMON ERRORS
• Rigid motions preserve distance.
• Rigid motions preserve collinearity.
• Rigid motions preserve angle measure.
• Rigid motions preserve parallelism.
• Rigid motions preserve betweenness.
Sometimes students recognize that a rigid motion is a rotation, but do not correctly identify the angle of rotation. Review the angles by having students stand and physically turn to show various rotations, such as 90° clockwise or 180° counterclockwise.
If a figure is determined by certain points, then its image after a rigid motion is determined by the images of those points. This is true because of the betweenness and collinearity properties of rigid motions. For example, suppose △ABC is determined by its vertices, points A, B, and C. You can find the image of △ABC by finding the images of points A, B, and C and connecting them with segments. Example 1
QUESTIONING STRATEGIES
Use coordinate notation to write the rule that maps each preimage to its image. Then identify the transformation and confirm that it preserves length and angle measure. B'
To write a rule for a rigid motion transformation, does it matter which point on the figure you choose? Explain. No, the same rule is applied to each point to transform the figure.
Preimage A(1, 2) B(4, 2) C(3, −2)
→ → →
Image A′(−2, 1) B′(−2, 4) C′(2, 3)
5
y C' A
-5
Look for a pattern in the coordinates.
A'
B
0
x 5
C
The x-coordinate of each image point is the opposite of the y-coordinate of its preimage.
-5
The y-coordinate of each image point equals the x-coordinate of its preimage.
The transformation is a rotation of 90° counterclockwise around the origin given by the rule (x, y) → (−y, x).
© Houghton Mifflin Harcourt Publishing Company
How does looking for a pattern in the coordinates from the preimage to the image help you identify the type of rigid motion? If the coordinates are changed by the same sum or difference, the rigid motion is a translation. If one coordinate stays the same and the other is the opposite of its original, the rigid motion is a reflection. If both coordinates change signs or if the x-and y-coordinates switch and one changes sign, the rigid motion is a rotation.
Describing Rigid Motions Using Coordinate Notation
Explain 1
Find the length of each side of △ ABC and △A′B′C′. Use the Distance Formula as needed. AB = 3
―――――――― BC = √(3 − 4) + (-2 − 2) = √― 17 ―――――――― AC = √(3 − 1) + (-2 − 2) = √― 20 2
2
2
2
A′ B′ = 3 B′ C′ = =
―――――――― + (3 − 4)
√(2 − (-2))
2
A′ C′ = √(2 − (-2)) + (3 − 1) 2
=
2
_ √ 17 ____
2
_ √ 20
Since AB = A′ B′ , BC = B′ C′ , and AC = A′ C′ , the transformation preserves length. Find the measure of each angle of △ABC and △A′B′C′. Use a protractor. m∠A = 63°, m∠B = 76°, m∠C = 41°
m∠A′ = 63°, m∠B′ = 76°, m∠C′ = 41°
Since m∠A = m∠A′ , m∠B = m∠B′ , and m∠C = m∠C′ , the transformation preserves angle measure. Module 16
803
Lesson 3
COLLABORATIVE LEARNING ACTIVITY IN1_MNLESE389762_U7M16L3 803
Small Group Activity Give each student three sheets of graph paper. Have each student draw a preimage and an image showing a translation on one sheet, a reflection on another, and a rotation on the third. Have them write the rule for each transformation on the back. Then, working in groups of three or four, have students analyze each other’s graphs and determine the rule used to create each transformation.
803
Lesson 16.3
4/19/14 11:57 AM
B
Preimage P(-3, -1) → Q(3, -1) → R(1, −4) →
Image P′(−3, 1) Q′(3, 1) R′(1, 4)
5
Look for a pattern in the coordinates.
-5
The x-coordinate of each image point equals the x-coordinate of its preimage.
P' P
R'
Q' Q
0
The y-coordinate of each image point is the opposite of the y-coordinate of its preimage.
-5
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Some students have difficulty visualizing
y
x 5
rotations. Have students copy the image on graph paper and then rotate the paper a given number of degrees. Tell them that the location and orientation of the figure on the graph after rotating the paper shows how the final image looks when the rotation rule is applied to the preimage.
R
reflection across the x-axis The transformation is a x, y ) → (x, -y) . ( given by the rule Find the length of each side of △PQR and △P′Q′R′. P′ Q′ = 6
PQ = 6 QR =
――――――――――
√(1 − 3 ) + (−4 − -1 ) 2
2
Q′ R′ =
――
2
2
2
√ 13 ―――――――――― P′ R′ = √( 1 − -3 ) + ( 4 − 1 ) ―― = √ 25 = 5
√ 13 ―――――――――― PR = √( 1 − -3 ) + ( −4 − -1 ) ―― = √ 25 = 5 =
―――――――――― 3 ) + (4 − 1 ) ――
√(1 −
=
2
2
2
Since PQ = P' Q' , QR = Q' R' , and PR = P' R' , the transformation preserves length. Find the measure of each angle of △PQR and △P′Q′R′. Use a protractor. m∠P ' = 37° , m∠Q' = 56° , m∠R' = 87°
Since m∠P = m∠P' , m∠Q = m∠Q' , and m∠R = m∠R' , the transformation preserves angle measure. Reflect
3.
How could you use a compass to test whether corresponding lengths in a preimage and image are the same? Place the point of the compass on one endpoint of the segment in the preimage and open it
to the length of the segment. Without adjusting the compass, move the point of the compass to an endpoint of the corresponding segment in the image and make an arc. If the lengths
© Houghton Mifflin Harcourt Publishing Company
m∠P = 37° , m∠Q = 56° , m∠R = 87°
are the same, the arc will pass through the other endpoint of the segment in the image. 4.
Look back at the transformations in the Explore. Classify each transformation as a rigid motion or not a rigid motion. A. rigid motion; B. rigid motion; C. not a rigid motion
Module 16
804
Lesson 3
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U7M16L3 804
Multiple Representations
4/19/14 11:57 AM
For each of the transformations, ask students to summarize all the possible ways to represent the transformation: graphically, using a rule, and using words. When students use words, encourage them to be as specific as possible, using the appropriate vocabulary from this lesson. Discuss the different advantages of each representation.
Representing and Describing Transformations
804
Your Turn
EXPLAIN 2
Use coordinate notation to write the rule that maps each preimage to its image. Then identify the transformation and confirm that it preserves length and angle measure.
Describing Nonrigid Motions Using Coordinate Notation
5.
Preimage D(-4, 4) E(2, 4) F(−4, 1)
Image D′(4, -4) E′(-2, -4) F′(4, -1)
→ → →
Each coordinate maps to its opposite.
CONNECT VOCABULARY
The transformation is a rotation of 180° around -5 the origin given by the rule (x, y) → (−x, -y). m∠D = m∠D′ = 90° DE = D′ E′ = 6
Note that the prefix pre- in preimage indicates “before,” and that the preimage is before the image in a transformation.
―
EF = E′ F′ = √45
DF = D'F' = 3
5
D
y E
x
F 0
m∠E = m∠E′ = 27°
5 F'
E' -5
m∠F = m∠F′ = 63°
D'
The transformation preserves length and angle measure. 6.
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 You may want to give students verbal
Image S′(-2, 2) T′(3, 2) U′(-1, -2)
→ → →
5
S S'
y T T'
x-coordinates: image is 1 more than preimage y-coordinates: image is 2 less than preimage
x -5
The transformation is a translation given by the rule (x, y) → (x + 1, y - 2). ST = S′ T′ = 5 TU = T′ U′ = √32
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mary Hockenbery/Flickr/Getty Images
descriptions of a variety of transformations. For example, the simplest transformation maps every point to itself. This is known as the identify transformation. Another transformation maps every point to the origin. Giving students examples of these “extreme” transformations will help them realize that not every transformation is a rigid motion.
Preimage S(-3, 4) T(2, 4) U(−2, 0)
― ― SU = S′ U′ = √17
m∠S = m∠S′ = 76°
m∠T = m∠T′ = 45°
U 0
5
U' -5
m∠U = m∠U′ = 59°
The transformation preserves length and angle measure.
Explain 2
Describing Nonrigid Motions Using Coordinate Notation
Transformations that stretch or compress figures are not rigid motions because they do not preserve distance. The view in the fun house mirror is an example of a vertical stretch.
Module 16
805
Lesson 3
LANGUAGE SUPPORT IN1_MNLESE389762_U7M16L3 805
Vocabulary Development After reading the properties of rigid motions, discuss the meaning of “preserving” the characteristics of rigid motions. Have students explain in their own words what it means for each property to be preserved.
805
Lesson 16.3
4/19/14 11:57 AM
Example 2
Use coordinate notation to write the rule that maps each preimage to its image. Then confirm that the transformation is not a rigid motion.
QUESTIONING STRATEGIES How can you use an equation or write a ratio to find the multiplicative factor in a nonrigid motion? Possible answer: If the corresponding coordinates of the preimage and image are a and b, solve the equation ax = b for x or use the ratio.
△JKL maps to triangle △J′K′L′. Image
Preimage J(4, 1)
→
J′(4, 3)
L(0, -3)
→
L′(0, -9)
K(-2, -1) →
K′(-2, -3)
image coordinate _ba or __ .
Look for a pattern in the coordinates.
preimage coordinate
The x-coordinate of each image point equals the x-coordinate of its preimage. The y-coordinate of each image point is 3 times the y-coordinate of its preimage. The transformation is given by the rule (x, y) → (x, 3y).
How can you use the rule for a nonrigid motion to recognize if a figure is stretched or compressed? If the absolute value of the multiplicative factor is greater than 1, the figure is stretched. If it is less than 1, the figure is compressed.
Compare the length of a segment of the preimage to the length of the corresponding segment of the image. ____ ――――――― 2 2 2 2 JK = √(-2 − 4) + (-1 − 1) J′ K′ = (-2 − 4) + (-3 − 3)
―
= √40
=
_ √ 72
Since JK ≠ J′K′ , the transformation is not a rigid motion.
Can a nonrigid motion stretch a figure in one direction and compress it in another direction? Explain. No, because it preserves the shape of the figure.
△ MNP maps to triangle △ M′ N′ P′. Preimage M(-2, 2)
→
P(-2, -2)
→
N(4, 0)
→
Image
M′(-4, 1) N′(8, 0)
P′(-4, -1)
The x-coordinate of each image point is twice the x-coordinate of its preimage. The y-coordinate of each image point is half the y-coordinate of its preimage.
(
)
1 x, y) → 2x, __ y 2 .
Compare the length of a segment of the preimage to the length of the corresponding segment of the image.
―――――――― ―――――――――― = √( 4 − -2 ) + ( 0 − 2 ) ――――― = √ 6 + -2 ――
MN =
√(x 2 − x 1) 2 + (y 2 − y 1) 2 2
2
=
Since
―――――――― ―――――――――――― = √( 8 − -4 ) + ( 0 − 1 ) ――――― = √ 12 + -1 ――
M′ N′ =
2
2
2
2
√ 40
=
MN ≠ M′ N′ , the transformation is not a rigid motion.
Module 16
IN1_MNLESE389762_U7M16L3 806
√(x 2 − x 1) 2 + (y 2 − y 1) 2
806
2
2
√145
© Houghton Mifflin Harcourt Publishing Company
The transformation is given by the rule(
Lesson 3
4/19/14 11:56 AM
Representing and Describing Transformations
806
Reflect
ELABORATE
7.
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Discuss with students why angle measure is
How could you confirm that a transformation is not a rigid motion by using a protractor? If any angle measure in the preimage is different from the corresponding angle measure in
the image, then the transformation is not a rigid motion. Therefore, use the protractor to check corresponding angles. If all angle measures are preserved, then check lengths. Your Turn
Use coordinate notation to write the rule that maps each preimage to its image. Then confirm that the transformation is not a rigid motion.
preserved but length is not preserved in a nonrigid motion.
8.
△ ABC maps to triangle △ A′ B′ C′. Preimage
AVOID COMMON ERRORS
→
→
Preimage
B′(6, 3)
S(4, 2)
A′(3, 3) C′(3, -6)
Image
R(-2, 1)
→
R′(-1, 3)
T(2, -2)
→
T ′(1, -6)
(
→
1 x, 3y (x, y) → _ 2
)
S′(2, 6)
―――――――― 37 RS = √(4 − (-2)) +( 2 − 1) = √― ―――――――― 18 R′ S′ = √(2 − (-1)) + (6 − 3) = √―
AB is horizontal and AB = 2. _ A′ B′ is horizontal and A′B′ = 3.
Since AB ≠ A′ B′, the transformation is not a rigid motion.
2
2
2
2
Since RS ≠ R′ S′, the transformation is not a rigid motion.
Elaborate
10. Critical Thinking To confirm that a transformation is not a rigid motion, do you have to check the length of every segment of the preimage and the length of every segment of the image? Why or why not? No; once you find a segment of the preimage whose length is not equal to the length of
the corresponding segment of the image, you can stop checking lengths. You only need © Houghton Mifflin Harcourt Publishing Company
How are transformations of nonrigid motions similar to and different from transformations of rigid motions? Rigid and nonrigid motions preserve angle measures, betweenness, and collinearity. Rigid motions preserve distance while nonrigid motions do not preserve distance.
C(2, -4)
△ RST maps to triangle △ R′S′ T ′ .
Image
x, y) → (1.5x, 1.5y) (_
CRITICAL THINKING
SUMMARIZE THE LESSON
→
B(4, 2)
Make sure students understand that both coordinates are multiplied by the same factor to create a nonrigid motion transformation.
Have students think about evaluating a classmate’s work to check if the properties of a rigid motion transformation have been correctly verified. Discuss what students should look for as they assess the work. Repeat for the properties of a nonrigid motion transformation
A(2, 2)
9.
to find one pair whose lengths are not equal in order to confirm that the transformation is not a rigid motion. 11. Make a Conjecture A polygon is transformed by a rigid motion. How are the perimeters of the preimage polygon and the image polygon related? Explain. The perimeters are equal. Each side of the preimage polygon is transformed to a side of
the image polygon with the same length. The sum of the side lengths of the preimage is equal to the sum of the side lengths of the image. 12. Essential Question Check-In How is coordinate notation for a transformation, such as (x, y) → (x + 1, y - 1), similar to and different from algebraic function notation, such as ƒ(x) = 2x + 1? In both cases, the notation shows how an input is changed by the transformation or
function. In coordinate notation, the input is a point of the coordinate plane. In algebraic function notation, the input is a real number. Module 16
IN1_MNLESE389762_U7M16L3 807
807
Lesson 16.3
807
Lesson 3
4/19/14 11:56 AM
Evaluate: Homework and Practice
EVALUATE • Online Homework • Hints and Help • Extra Practice
Draw the image of each figure under the given transformation. Then describe the transformation in words. 1.
(x, y) → (−x, −y) 5
2.
(x, y) → (x + 5, y)
y A
Q
P C -5
C'
A(4, 3)
→ A′(-4, -3)
R'
P (-4, 2)
= A′(-4, -3)
→ B′(-3, -1)
= B′(-3, -1)
D(2, 4)
→ D′(-2, -4)
Q(-1, 3)
(
5
4.
F
E
( ( (
)
)
)
Module 16
Exercise
5 K'
M
1 ⋅3 → D′ 1, _ = D′ (1, 1) 3 = E′ (3, -1) E (3, -3) → E′ 3, 1 ⋅ -3 3 F (-3, -3) → F′ -3, 1 ⋅ -3 = F′ (-3, -1) 3 1 vertical compression by a factor of __ 3
_ _
0
M'
-5
L
-5
K (-2, 1)
L (4,-3)
→
K‘ (1, -2)
→ L‘ (-3, 4)
Explore Performing Transformations Using Coordinate Notation
Exercises 1–4
Example 1 Describing Rigid Motions Using Coordinate Notation
Exercises 5–6
Example 2 Describing Nonrigid Motions Using Coordinate Notation
Exercises 7–9
transformation from its rule before drawing the image. Then have them draw the image to verify their predictions and highlight any misconceptions.
M(-2, -4) → M‘ (-4, -2)
reflection across the line y = x
Lesson 3
808
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–4
1 Recall of Information
MP.4 Modeling
5–6
1 Recall of Information
MP.2 Reasoning
7–8
2 Skills/Concepts
MP.3 Logic
9
2 Skills/Concepts
MP.4 Modeling
1 Recall of Information
MP.4 Modeling
12
2 Skills/Concepts
MP.4 Modeling
13–14
2 Skills/Concepts
MP.4 Modeling
10–11
© Houghton Mifflin Harcourt Publishing Company
E'
x
-5
5
Practice
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Encourage students to try to identify the
y
K
x
F'
5
L'
0
= Q′(4, 3)
Concepts and Skills
(x, y) → (y, x)
y
D'
IN1_MNLESE389762_U7M16L3 808
→ Q′(-1 + 5, 3)
= P′(1, 2)
translation 5 units right
D
D(1, 3)
→ P′(-4 + 5, 2)
R(-3, -3) → R′(-3 + 5, -3) = R′(2, -3)
= D′(-2, -4)
rotation of 180° around the origin 1y (x, y) → x, _ 3 )
-5
5
-5
C(-2, 1) → C′(-(-2), -1) = C′(2, -1)
3.
0
D'-5
B(3, 1)
ASSIGNMENT GUIDE x
-5
5
R
A'
Q'
P'
x
B 0
B'
y
5
D
4/19/14 11:56 AM
Representing and Describing Transformations
808
Use coordinate notation to write the rule that maps each preimage to its image. Then identify the transformation and confirm that it preserves length and angle measure.
GRAPHIC ORGANIZERS Have students make a graphic organizer or table to compare properties of reflections, rotations, translations, and nonrigid motions.
5.
Preimage
Image
A(-4, 4) B(-1, 2) C(-4, 1)
A′(4, 4) B′ (2, 1) C′ (1, 4)
→ → →
― ― √ BC = B′C′ = 10
AVOID COMMON ERRORS Some students may draw an image and then incorrectly label the corresponding vertices of the image. Suggest that students label each vertex as they plot the point for the transformation.
C -5
m∠A = m∠A′ = 56° m∠C = m∠C′ = 72°
-5
Image J′ (-3, 0) K′(-3, -4) L′ (-1, -2)
→ → →
5
-5
m∠J = m∠J′ = 45°
― ―
KL = K′L′ = √8 JL = J′L′ = √8
y
J
(x, y) → (-y, -x); reflection across the line y = - x JK = J′K′ = 4
x 5
m∠B = m∠B′ = 52°
AC = A′C′ = 3
Preimage J(0, 3) K(4, 3) L(2, 1)
A'
B'
0
The transformation preserves length and angle measure. 6.
y C'
B
(x, y) → (y, -x); rotation of 90° clockwise around the origin AB = A′B′ = √13
5
A
K x
L
J'
0
m∠K = m∠K′ = 45°
5
L'
m∠L = m∠L′ = 90°
K'
-5
The transformation preserves length and angle measure. Use coordinate notation to write the rule that maps each preimage to its image. Then confirm that the transformation is not a rigid motion. 7.
△ABC maps to triangle △A′ B′C′.
© Houghton Mifflin Harcourt Publishing Company
Preimage
B′(2, -1)
G(1, -1)
→
A′(3, 3)
C(0, 0)
→
C′(0, 0)
(
→
)
F(-1, 1)
2
2
2
2
Module 16
Lesson 16.3
G′(2, -1)
H′(-4, -2)
2
2
2
2
Since FG ≠ F′ G′, the transformation is not a rigid motion.
Lesson 3
809
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
3 Strategic Thinking
MP.3 Logic
17
2 Skills/Concepts
MP.3 Logic
18
2 Skills/Concepts
MP.6 Precision
19
3 Strategic Thinking
MP.2 Reasoning
20
3 Strategic Thinking
MP.6 Precision
15–16
Image
F′(-2, 1)
――――――――― √(1 - (-1)) + (-1 - 1) = √― 8 ――――――――― F′ G′ = √(2 - (-2)) + (-1 - 1) = √― 20 FG =
Since AB ≠ A′ B′, the transformation is not a rigid motion.
809
→
(x, y) → (2x, y)
―――――――― AB = √(4 - 6) + (-2 - 6) = √― 68 ―――――――― A′ B′ = √(2 - 3) + (-1 - 3) = √― 17
Exercise
→
H(-2, -2) →
1y 1 x, _ (x, y) → _ 2 2
IN1_MNLESE389762_U7M16L3 809
△FGH maps to triangle △F′ G′ H′. Preimage
A(6, 6)
B(4, -2)
8.
Image
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9.
Analyze Relationships A mineralogist is studying a quartz crystal. She uses a computer program to draw a side view of the crystal, as shown. She decides to make the drawing 50% wider, but to keep the same height. Draw the transformed view of the crystal. Then write a rule for the transformation using coordinate notation. Check your rule using the original coordinates. y F'
A A' F
B B' x
-5
0 E'
E
(x, y) → (1.5x, y)
A(0, 4) B(2, 3)
→ A′(1.5 ⋅ 0, 4) → B′(1.5 ⋅ 2, 3)
C(2, -3) → C′(1.5 ⋅ 2, -3) D(0, -4) → D′(1.5 ⋅ 0, -4)
5 C
D D'
C'
= A′(0, 4) = B′(3, 3)
= C′(3, -3) = D′(0, -4)
E(-2, -3) → E′(1.5 ⋅ -2, -3) = E′(-3, -3) F(-2, 3)
→ F′(1.5 ⋅ -2, 3)
= F′(-3, 3)
10. Use the points A(2, 3) and B(2, -3).
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Charles D. Winters/Photo Researchers/Getty Images
a. Describe segment AB and find its length. Segment AB is a vertical segment that is 6 units long. b. Describe the image of segment AB under the transformation (x, y) → (x, 2y). A(2, 3) → A′(2, 2 ⋅ 3) = A′(2, 6)
B(2, -3) → B′(2, 2 ⋅ (-3)) = B′(2, -6)
c.
The image of segment AB is a vertical segment that is 12 units long. Describe the image of segment AB under the transformation (x, y) → (x + 2, y). A(2, 3)
→ A′(2 + 2, 3)
= A′(4, 3)
B(2, -3) → B′(2 + 2, -3) = B′(4, -3) The image of segment AB is a vertical segment two units to the right of the original segment that is 6 units long. d. Compare the two transformations. Possible answer: (x, y) → (x + 2, y) is rigid, because it does not change the length of the segment. (x, y) → (x, 2y) is not rigid because it doubles the length of the segment. The segment remains vertical under both transformations.
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Representing and Describing Transformations
810
11. Use the points H(-4, 1) and K (4, 1).
CRITICAL THINKING
a. Describe segment HK and find its length. Segment HK is a horizontal segment that is 8 units long.
A reflection in the coordinate plane can be across either axis, any vertical or horizontal line, or any other line, such as y = x or y = -x. Discuss with students how to identify the line of reflection for a reflection.
b. Describe the image of segment HK under the transformation (x, y) → (-y, x). H(-4, 1) → H′(-1, -4)
K(4, 1) → K′(-1, 4) The image of segment HK is a vertical segment that is 8 units long.
Describe the image of segment HK under the transformation (x, y) → (2x, y). H(-4, 1) → H′(-8, 1)
c.
K(4, 1) → K′(8, 1) The image of segment HK is a horizontal segment that is 16 units long.
d. Compare the two transformations. Possible answer: (x, y) → (-y, x) is rigid, because it does not change the length of the segment. (x, y) → (2x, y) is not rigid because it doubles the length of the segment. The transformation given by (x, y) → (-y, x) switches the segment from horizontal to vertical, while (x, y) → (2x, y) does not.
12. Make a Prediction A landscape architect designs a flower bed that is a quadrilateral, as shown in the figure. The plans call for a light to be placed at the midpoint of the longest side of the flower bed. The architect decides to change the location of the flower bed using the transformation (x, y) → (x, -y). Describe the location of the light in the transformed flower bed. Then make the required calculations to show that your prediction is correct. 5
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Andrew Fletcher/Shutterstock
A
y B C x
D -5
0
5
-5
_ The longest side of ABCD is CD and its midpoint is
(
)
4 + (-2) 3 + 1 M _, _ = M(1, 2). 2 2 The coordinates of the transformed flower bed are: A(-3, 3) → A′(-3, -3)
C(4, 3)
→ C′(4, -3)
M(1, 2) → M′(1, -2)
D(-2, 1) → D′(-2, -1) _ The longest side of the transformed flower bed is C'D' and its midpoint is B(1, 4)
(
→ B′(1, -4)
)
4 + (-2) -3 + (-1) M′ _, __ = M′(1, -2). 2 2 This matches the coordinates of the image of M calculated above, so the prediction is correct.
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13. Multiple Representations If a transformation moves points only up or down, how do the coordinates of the point change? What can you conclude about the coordinate notation for the transformation?
COMMUNICATING MATH Have students work in pairs. One student draws a figure in a coordinate plane, and gives clues to the second student for drawing a second image. A sample clue: “Draw this pre-image transformed into a mirror image, or reflection.” The second student draws the image, and then explains the reason the image is a reflection. They switch roles; the second student draws a pre-image, and asks the first student to draw a rotated image. The first student has to explain how the image drawn fulfills the clue.
The x-coordinate does not change. The y-coordinate has a constant added to it (for a translation up) or subtracted from it (translation down). The coordinate notation has the form (x, y) → (x, y + b) , where b is a real number, b ≠ 0.
14. Match each transformation with the correct description. E dilation with scale factor 3 A. x, y → (3x, y)
(
)
B. (x, y) → (x + 3, y)
D
translation 3 units up
C. (x, y) → (x, 3y)
B
translation 3 units right
D. (x, y) → (x, y + 3)
A
horizontal stretch by a factor of 3
E. (x, y) → (3x, 3y)
C
vertical stretch by a factor of 3
MODELING
Draw the image of each figure under the given transformation. Then describe the transformation as a rigid motion or not a rigid motion. Justify your answer. 15. (x, y) → (2x + 4, y)
P(-2, 3)
→ P′(2(-2) + 4, 3)
= P′(0, 3)
P
Q(-1, -3) → Q′(2(-1) + 4, -3) = Q′(2, -3)
R(-3, -3) → R′(2(-3) + 4, -3) = R′(-2, -3)
QR = 2
Q′R′ = 4
0 R R'
DE = 6
→ E′(0.5 ⋅ 2, 3 - 4)
Q
Q'
5
D(-4, 3) → D′(0.5 ⋅ -4, 3 - 4) = D′(-2, -1) → F′(0.5 ⋅ 4, 1 - 4)
5
D
E
= E′(1, -1)
= F′(2, -3)
F -5
D'
0
D′E′ = 3
Since DE ≠ D′E′, the transformation is not a rigid motion.
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y
812
E'
x 5
SMALL GROUP ACTIVITY
© Houghton Mifflin Harcourt Publishing Company
16. (x, y) → (0.5x, y - 4)
F(4, 1)
P' x
-5
Since QR ≠ Q′R′, the transformation is not a rigid motion.
E(2, 3)
If the coordinates of a transformation have different multiplicative factors, the transformation is neither a rigid motion nor a nonrigid motion. Discuss why this is the case and how to use the multiplicative factor to recognize whether the transformation is a vertical or horizontal stretch or compression.
y
Is reflecting a figure across the x-axis and then across the y-axis the same as rotating it 180°? Have students work in small groups to explore the question both by drawing examples on the coordinate plane and by writing and analyzing the algebraic rules for the transformations. When they have had time to formulate an answer, have each group present their conclusion and the reasoning behind it.
F' -5
Lesson 3
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Representing and Describing Transformations
812
JOURNAL
H.O.T. Focus on Higher Order Thinking
Have students explain and give an example on a coordinate plane of a reflection, a translation, a rotation, and a nonrigid motion transformation. Have them write the rules for their translations and nonrigid motion transformations.
17. Explain the Error A student claimed that the transformation (x, y) → (3x, y) is a rigid motion because the segment joining (5, 0) to (5, 2) is transformed to the segment joining (15, 0) to (15, 2), and both of these segments have the same length. Explain the student’s error.
The transformation is a horizontal stretch by a factor of 3, so it preserves the length of vertical segments but not the length of horizontal or diagonal segments. In order to be a rigid motion, the transformation must preserve all lengths, so this transformation is not a rigid motion.
18. Critical Thinking Write a rule for a transformation that maps △STU to △S′T′U′.
S(2, 3)
T(4, -3)
y
5
S' S
→ S′(1, 3)
→ T′(2, -3)
U(-4, -3) → U′(-2, -3)
x
-5
0
5
So, the transformation divides each x-coordinate by 2 but leaves the y-coordinate unchanged. 1 The rule is (x, y) → x, y . 2
U
U'
(_ )
T'
T
-5
© Houghton Mifflin Harcourt Publishing Company
19. Justify Reasoning Consider the transformation given by the rule (x, y) → (0, 0). Describe the transformation in words. Then explain whether or not the transformation is a rigid motion and justify your reasoning.
The transformation maps all points to the origin, so the image of any figure under this transformation is a single point, (0, 0). The transformation is not a rigid motion because all line segments are mapped to a point, so the length of the segment is not preserved.
20. Communicate Mathematical Ideas One of the properties of rigid motions states that rigid motions preserve parallelism. Explain what this means, and give an example using a specific figure and a specific rigid motion. Include a graph of the preimage and image.
If two segments or lines are parallel in the preimage, then the corresponding lines or segments of the image are also parallel. _ _ Possible example: In ABCD, AB∥CD. A rotation of 90° counterclockwise around _ the origin _ is a rigid motion, and after this transformation A′B′∥C′D′.
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B'
-5
A'
5 C'
y A
D' D 0
B C x 5
-5
Lesson 3
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Lesson Performance Task
INTEGRATE TECHNOLOGY Lengths and widths on a computer screen are measured in pixels. A pixel is the smallest visual element that a computer is capable of processing. Therefore, it is always a whole number. Typical dimensions for a smartphone portrait are 310 pixels by 352 pixels, and for a large computer screen 1024 pixels by 768 pixels.
A Web designer has created the logo shown here for Matrix Engineers.
Matrix Engineers The logo is 100 pixels wide and 24 pixels high. Images placed in Web pages can be stretched horizontally and vertically by changing the dimensions in the code for the Web page.
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Ask students to visualize a rectangular piece
The Web designer would like to change the dimensions of the logo so that lengths are increased or decreased but angle measures are preserved.
a. Find three different possible sets of dimensions for the width and height so that lengths are changed but angle measures are preserved. The dimensions must be whole numbers of pixels. Justify your choices.
of rubber that they are holding between their hands. In the middle is a large plus sign. Have them describe what happens to the plus sign in each of the following situations.
b. Explain how the Web designer can use transformations to find additional possible dimensions for the logo.
a. Possible dimensions: 150 pixels by 36 pixels, 200 pixels by 48 pixels, 250 pixels by 60 pixels A horizontal stretch changes lengths but does not preserve angle measures; a vertical stretch changes lengths but does not preserve angle measures. However, stretching both horizontally and vertically by the same scale factor (a dilation) changes lengths while preserving angle measures. The dimensions given represent dilations by a scale factor of 1.5, 2, and 2.5, respectively.
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b. In general, the required dimensions should be 100k pixels and 24k pixels, where k is any real number, k > 0, such that 100k and 24k are whole numbers. The Web designer should transform the logo using a dilation, (x, y) → (kx, ky), where k > 0 and such that 100k and 24k are whole numbers.
• They stretch the rubber horizontally by moving their hands farther apart. The horizontal bar of the plus sign increases in length. The vertical bar stays the same height. • They stretch the rubber vertically by grabbing it at the top and bottom and moving their hands farther apart. The vertical bar of the plus sign increases in length. The horizontal bar stays the same width. • Together with a friend, they stretch the rubber to an equal extent both vertically and horizontally. Both bars of the plus sign increase in length and the lengths increase proportionally. The result is that the plus sign gets larger but still looks like a plus sign.
Lesson 3
EXTENSION ACTIVITY IN1_MNLESE389762_U7M16L3 814
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Have students plot the points (5, 5), (8, 5), (8, 8), and (5, 8) on a coordinate grid. They should connect the points to form a square and draw the diagonals of the square. Ask them to experiment now by drawing on the same grid: (a) rectangles with heights of 3 units, lengths other than 3 units, and diagonals; and (b) rectangles with lengths of 3 units, heights other than 3 units, and diagonals. The rectangles are the square stretched or compressed horizontally or vertically. For each rectangle, students should compare the angles formed by the diagonals with the angles of the diagonals of the original square.
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Representing and Describing Transformations
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LESSON
16.4
Name
Reasoning and Proof
Class
Date
16.4 Reasoning and Proof Essential Question: How do you go about proving a statement?
Common Core Math Standards The student is expected to: COMMON CORE
Resource Locker
G-CO.C.9
Explore
Prove theorems about lines and angles. Also A-REI.A.1
A conjecture is a statement that is believed to be true. You can use inductive or deductive reasoning to show, or prove, that a conjecture is true. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. Deductive reasoning is the process of using logic to draw conclusions.
Mathematical Practices COMMON CORE
MP.8 Patterns
Language Objective
Complete the steps to make a conjecture about the sum of three consecutive counting numbers.
Have students fill in a chart explaining the meaning of conditionals, counterexample and conditional.
ENGAGE
Write a sum to represent the first three consecutive counting numbers, starting with 1.
Is the sum divisible by 3?
Possible answer: You can make a conjecture, or statement, that you believe is true. Then through inductive or deductive reasoning, you can prove the statement is true by showing specific cases are true or by using logical steps.
View the online Engage. Discuss the photo. Discuss why the figure seems to be impossible. Then preview the Lesson Performance Task.
© Houghton Mifflin Harcourt Publishing Company
Essential Question: How do you go about proving a statement?
PREVIEW: LESSON PERFORMANCE TASK
Exploring Inductive and Deductive Reasoning
1+2+3 Yes. 1 + 2 + 3 = 6 and 6 ÷ 3 = 2.
Write the sum of the next three consecutive counting numbers, starting with 2.
2+ 3+4
Is the sum divisible by 3?
Yes. 2 + 3 + 4 = 9 and 9 ÷ 3 = 3.
Complete the conjecture: The
sum
of three consecutive counting numbers is divisible by
3
.
Recall that postulates are statements you accept are true. A theorem is a statement that you can prove is true using a series of logical steps. The steps of deductive reasoning involve using appropriate undefined words, defined words, mathematical relationships, postulates, or other previously-proven theorems to prove that the theorem is true.
Use deductive reasoning to prove that the sum of three consecutive counting numbers is divisible by 3.
Let the three consecutive counting numbers be represented by n, n + 1, and
The sum of the three consecutive counting numbers can be written as 3n +
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Lesson 4 815 Module 16
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(
n+1
).
H
The expression 3n + 3 can be factored as 3
I
The expression 3(n + 1) is divisible by
J
Recall the conjecture in Step E: The sum of three consecutive counting numbers is divisible by 3.
EXPLORE
for all values of n.
3
Look at the steps in your deductive reasoning. Is the conjecture true or false?
Exploring Inductive and Deductive Reasoning True
QUESTIONING STRATEGIES
Reflect
1.
When a detective solves a case, is the detective more likely to use inductive or deductive reasoning? Explain. deductive reasoning, because the solution is likely based on logical conclusions drawn from the evidence
Discussion A counterexample is an example that shows a conjecture to be false. Do you think that counterexamples are used mainly in inductive reasoning or in deductive reasoning? Possible answer: A counterexample would be used in inductive reasoning to show that at least one specific case makes the conjecture false.
2.
Suppose you use deductive reasoning to show that an angle is not acute. Can you conclude that the angle is obtuse? Explain. No; if the angle is not acute, I can conclude that it is right, obtuse, or straight.
Explain 1
When might you want to make a conjecture about a set of numbers? If the numbers seem to form a pattern, you might want to make a conjecture based on the number pattern. Is one counterexample enough to prove that a conjecture is false? Explain. Yes, the conjecture must be true for every case. So, if even one counterexample exists, the conjecture is false.
Introducing Proofs
A conditional statement is a statement that can be written in the form “If p, then q” where p is the hypothesis and q is the conclusion. For example, in the conditional statement “If 3x - 5 = 13, then x = 6,” the hypothesis is “3x - 5 = 13” and the conclusion is “x = 6 .”
Properties of Equality Addition Property of Equality
If a = b, then a + c = b + c.
Subtraction Property of Equality
If a = b, then a - c = b - c.
Multiplication Property of Equality
If a = b, then ac = bc.
Division Property of Equality
If a = b and c ≠ 0, then __ac = _bc .
Reflexive Property of Equality
a=a
Symmetric Property of Equality
If a = b, then b = a.
Transitive Property of Equality
If a = b and b = c, then a = c.
Substitution Property of Equality
If a = b, then b can be substituted for a in any expression.
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Most of the Properties of Equality can be written as conditional statements. You can use these properties to solve an equation like “3x - 5 = 13” to prove that “x = 6 .”
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss why a conjecture is like a hypothesis in the scientific method. Elicit that a conjecture, like a hypothesis, is often based on inductive reasoning.
COMMUNICATING MATH Have students write a conjecture about numbers and then use examples to determine whether it is true. Lesson 4
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M16L4 816
Integrate Mathematical Practices
4/19/14 12:06 PM
This lesson provides an opportunity to address Mathematical Practice MP.3, which calls for students to “construct viable arguments.” Students use deductive reasoning, and explain steps logically from definite premises to a definite general conclusion. They use inductive reasoning to make a conjecture about what is true in general by examining several cases, and they justify the falsehood of a conclusion by citing a counterexample.
Reasoning and Proof 816
Use deductive reasoning to solve the equation. Use the Properties of Equality to justify each step.
Example 1
EXPLAIN 1
14 = 3x - 4
Introducing Proofs
14 = 3x - 4 18 = 3x
Addition Property of Equality
AVOID COMMON ERRORS
6=x
Division Property of Equality
Students may have difficulty identifying the correct property of equality to justify a step when solving an algebraic equation. Students may need to include steps where they show the property of equality to help them recognize how it is applied. For example, they may need to show the step where the value is added to both sides of the equation to apply the Addition Property of Equality.
x= 6
Symmetric Property of Equality
9 = 17 - 4x
-8
= -4x
Division
=x
2
x=
2
Property of Equality
Symmetric Property of Equality
Your Turn
Write each statement as a conditional. 3. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Digital Vision/Getty Images
Do you always use deductive reasoning when you solve an equation algebraically? Explain. Yes, you should always be able to support each step in the solution using a property.
Subtraction Property of Equality
9 - 17 = -4x
QUESTIONING STRATEGIES Will changing the order of the hypothesis and the conclusion in a true conditional statement change whether or not the statement is true? Explain. Yes, the statement may still be true but you would have to prove that it is.
9 = 17 - 4x
4.
All zebras belong to the genus Equus. If an animal is a zebra, then it belongs to the genus Equus.
The bill will pass if it gets two-thirds of the vote in the Senate.
If the bill gets two-thirds of the vote in the Senate, then it will pass. 5.
Use deductive reasoning to solve the equation 3 - 4x = -5.
3 - 4x = -5 -4x = -8 x=2 6.
Subtraction Property of Equality Division Property of Equality
Identify the Property of Equality that is used in each statement. If x = 2, then 2x = 4.
Multiplication Property of Equality
5 = 3a; therefore, 3a = 5 .
Symmetric Property of Equality
If T = 4, then 5T + 7 equals 27.
Substitution Property of Equality
If 9 = 4x and 4x = m, then 9 = m.
Transitive Property of Equality
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Small Group Activity Have students work in small groups. The first student writes a number or draws a shape. The next student writes or draws a second item, beginning a pattern. Have them continue until each student has contributed to the pattern. Then ask the first student to describe a rule for the pattern. Have the groups repeat this activity until each student has gone first.
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Explain 2
Using Postulates about Segments and Angles
EXPLAIN 2
Recall that two angles whose measures add up to 180° are called supplementary angles. The following theorem shows one type of supplementary angle pair, called a linear pair. A linear pair is a pair of adjacent angles whose noncommon sides are opposite rays. You will prove this theorem in an exercise in this lesson.
Using Postulates about Segments and Angles
The Linear Pair Theorem If two angles form a linear pair, then they are supplementary.
4
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Have students identify the property of
3
m∠3 + m∠4 = 180°
equality used as they complete each step to solve for the variable using the postulates about segments and angles.
You can use the Linear Pair Theorem, as well as the Segment Addition Postulate and Angle Addition Postulate, to find missing values in expressions for segment lengths and angle measures. Example 2
Use a postulate or theorem to find the value of x in each figure.
Given: RT = 5x - 12 x+ 2 R
QUESTIONING STRATEGIES 3x - 8
S 5x - 12
How are the segment and angle addition postulates applied to solve for a variable? Set the sum of the non-overlapping segments or angles equal to the measure of the whole segment or angle.
T
Use the Segment Addition Postulate. RS + ST = RT
(x + 2) + (3x - 8) = 5x - 12 6=x x=6
Module 16
AVOID COMMON ERRORS
© Houghton Mifflin Harcourt Publishing Company
4x - 6 = 5x - 12
818
When solving equations using the Segment or Angle Addition Postulates, students may forget to combine like terms or use inverse operations to solve. Review how to combine like terms and use inverse operations as needed.
Lesson 4
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U7M16L4 818
Kinesthetic Experience
4/19/14 12:06 PM
Have students act out the Reflexive, Symmetric, and Transitive Properties. For the Reflexive Property, have students look in a mirror. For the Symmetric Property, have two students stand next to each other and then change places. For the Transitive Property, have one student give a second student a sheet of paper, and have the second student give the paper to a third. The result is the same as if the first student had given the paper directly to the third.
Reasoning and Proof 818
Given: m∠RST = (5x + 6)°
B
R P
(x + 25)°
(5x + 10)°
S
T
Use the Angle Addition Postulate. m∠RST = m∠
RSP
+ m∠
PST
(15x − 10)° = (x + 25) ° + (5x + 10)° 15x − 10 = (6x + 35)
9
x=
45
x=
5
Reflect
7.
Discussion The Linear Pair Theorem uses the terms opposite rays as well as adjacent angles. Write a definition for each of these terms. Compare your definitions with your classmates. Possible answers: Opposite rays are rays that share a common endpoint and form a line.
© Houghton Mifflin Harcourt Publishing Company
Adjacent angles are two angles in the same plane with a common vertex and a common side, but no common interior points. Your Turn
8.
Two angles LMN and NMP form a linear pair. The measure of ∠LMN is twice the measure of ∠NMP. Find m∠LMN.
Use the Linear Pair Theorem. Substitute for m∠LMN. m∠LMN + m∠NMP = 180°
(2 ⋅ m∠NMP) + m∠NMP = 180°
3 ⋅ m∠NMP = 180° m∠NMP = 60° m∠LMN = 2 ⋅ m∠NMP = 2 ⋅ 60° = 120°
Module 16
819
Lesson 4
LANGUAGE SUPPORT IN1_MNLESE389762_U7M16L4 819
Vocabulary Development For the properties of equality based on the operations, have students highlight the operation to connect to the corresponding property. For the Reflexive, Symmetric, and Transitive Properties, discuss what these words bring to mind. For example, reflexive might remind students of a reflection in a mirror. You see the same thing on both sides of a mirror, so, a = a.
819
Lesson 16.4
4/19/14 12:06 PM
Explain 3
Using Postulates about Lines and Planes
EXPLAIN 3
Postulates about points, lines, and planes help describe geometric figures.
Postulates about Points, Lines, and Planes
Using Postulates about Lines and Planes
Through any two points, there is exactly one line.
COOPERATIVE LEARNING Through any three noncollinear points, there is exactly one plane containing them.
To help students understand the rationale behind the postulates about points, lines, and planes, ask them to draw additional examples with the points and lines in different locations to demonstrate each postulate. Share the drawings with the class.
If two points lie in a plane, then the line containing those points lies in the plane.
If two lines intersect, then they intersect in exactly one point.
If two planes intersect, then they intersect in exactly one line. © Houghton Mifflin Harcourt Publishing Company
Module 16
IN1_MNLESE389762_U7M16L4 820
820
Lesson 4
4/19/14 12:06 PM
Reasoning and Proof 820
Example 3
QUESTIONING STRATEGIES
Use each figure to name the results described.
Must any two planes intersect? Why or why not? Name planes in the classroom that support your answer. No, if the planes are parallel they will never intersect. Possible example: opposite walls in the classroom
A B
E
C
D
If a line lies in a plane, how many points of intersection do the line and the plane have? an infinite number: every point that lies on the line
Description
VISUAL CUES Students may have difficulty interpreting the diagrams showing intersecting planes. Make a slit in the side of one piece of paper and hold the paper in a horizontal plane. Slide a second sheet in a vertical plane perpendicular to the horizontal plane to provide students with a visual demonstration of two intersecting planes. Locate points and lines as needed.
Example from the figure
the line of intersection of two planes
Possible answer: The two planes intersect in line BD.
the point of intersection of two lines
The line through point A and the line through point B intersect at point C.
three coplanar points
Possible answer: The points B, D, and E are coplanar.
three collinear points
The points B, C, and D are collinear.
m
F
H © Houghton Mifflin Harcourt Publishing Company
J
Description
Module 16
IN1_MNLESE389762_U7M16L4 821
821
Lesson 16.4
ℓ G
Example from the figure
the line of intersection of two planes
Possible answer: The two planes intersect in line JF.
the point of intersection of two lines
The line through point F and the line through point H intersect at point J.
three coplanar points
Possible answer: The points F, J, and H are coplanar.
three collinear points
The points F, J, and G are collinear.
821
Lesson 4
4/19/14 12:06 PM
Reflect
9.
ELABORATE
Find examples in your classroom that illustrate the postulates of lines, planes, and points. Possible answers: walls, floors, corners, desktops, blackboard
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Point out that students are starting to build a
10. Draw a diagram of a plane with three collinear points and three points that are noncollinear. Possible answer: in the figure, B, C, and A are collinear; D, C, and A are noncollinear. B D
C
A
catalog of definitions, postulates, and theorems about geometric relationships that they will use throughout the course. Discuss why it is important to become familiar with using them as tools in deductive reasoning.
Elaborate 11. What is the difference between a postulate and a definition? Give an example of each. Possible answers: A postulate is a statement that is self-evident or is generally accepted to
be a true statement. A definition is a statement that explains the meaning of a word in terms of previously accepted words or statements.
QUESTIONING STRATEGIES
Possible examples:
How can you use a Property of Equality to write the equation 4x = 8 as a conditional statement? Using the Division Property of Equality, x = 2. The conditional statement is “If 4x = 8, then x = 2.”
Postulate: x = x is called the Reflexive Property of Equality. Definition: An even number is a number that is divisible by 2. 12. Give an example of a diagram illustrating the Segment Addition Postulate. Write the Segment Addition Postulate as a conditional statement. R
S
T
If S is between R and T, then RS + ST = RT.
CONNECT VOCABULARY
© Houghton Mifflin Harcourt Publishing Company
13. Explain why photographers often use a tripod when taking pictures. Through any three noncollinear points,
there is only one plane, so the feet of the tripod are all always flat against the plane of the ground, which steadies the camera. 14. Essential Question Check-In What are some of the reasons you can give in proving a statement using deductive reasoning? You can use given facts, definitions, postulates or properties, and
Connect conditionals in geometry to conditional statements in everyday life. For example: “If I walk in the rain without an umbrella, then I will get wet.” Have students express their own conditional statements tied to reality so that they can see this connection.
previously-proven theorems.
Module 16
IN1_MNLESE389762_U7M16L4 822
SUMMARIZE THE LESSON
822
Lesson 4
4/19/14 12:06 PM
How can you use deductive reasoning to establish a conclusion? Provide a logical sequence of statements supported by postulates and established theorems in which each statement logically follows from the preceding statement up to the conclusion.
Reasoning and Proof 822
Evaluate: Homework and Practice
EVALUATE
Explain why the given conclusion uses inductive reasoning. 1.
Find the next term in the pattern: 3, 6, 9. The next term is 12 because the previous terms are multiples of 3.
• Online Homework • Hints and Help • Extra Practice
The conclusion is based on observing three numbers. 2.
ASSIGNMENT GUIDE
3 + 5 = 8 and 13 + 5 = 18, therefore the sum of two odd numbers is an even number.
The conclusion is based on two examples.
Concepts and Skills
Practice
Explore Exploring Inductive and Deductive Reasoning
Exercises 1–12
Example 1 Introducing Proofs
Exercises 13–16
Example 2 Using Postulates about Segments and Angles
Exercises 17–20
Example 3 Using Postulates about Lines and Planes
Exercises 21–27
3.
My neighbor has two cats and both cats have yellow eyes. Therefore when two cats live together, they will both have yellow eyes.
The conclusion is based on two observations. 4.
The conclusion is based on a limited number of observations. Give a counterexample for each conclusion. 5.
6.
Points A, B, and C are noncollinear, so therefore they are noncoplanar.
When I draw three points that are noncollinear, I can draw a single plane through all three points, so they are coplanar after all. © Houghton Mifflin Harcourt Publishing Company
Students work in pairs to complete a chart. The chart has three columns and each column is labeled with the highlighted vocabulary. Students must discuss in depth the meaning of each word, and take notes on their discussion. Then they write down their agreed-upon ideas under the word in each column.
The difference between two even numbers is positive.
Counterexample: 6 − 10 = −4, which is negative.
COMMUNICATING MATH
GRAPHIC ORGANIZERS
If x is a prime number, then x + 1 is not a prime number.
Counterexample: 2; if x = 2, then x + 1 = 3, which is a prime number.
7.
Have students compare counterexamples used to demonstrate conjectures that are not true.
It always seems to rain the day after July 4th.
8.
The square of a number is always greater than the number. 1 __ 1 The square of __ is 19 , which is less than __ . 3 3
In Exercises 9–12 use deductive reasoning to write a conclusion. 9.
If a number is divisible by 2, then it is even. The number 14 is divisible by 2.
The number 14 is an even number.
Module 16
Exercise
IN1_MNLESE389762_U7M16L4 823
Lesson 16.4
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–6
1 Recall of Information
MP.3 Logic
7–8
2 Skills/Concepts
MP.3 Logic
9–16
1 Recall of Information
MP.3 Logic
17–20
1 Recall of Information
MP.2 Reasoning
21–22
1 Recall of Information
MP.3 Logic
23–24
2 Skills/Concepts
MP.3 Logic
3 Strategic Thinking
MP.3 Logic
25
823
Lesson 4
823
4/19/14 12:06 PM
Use deductive reasoning to write a conclusion.
AVOID COMMON ERRORS
10. If two planes intersect, then they intersect in exactly one line. Planes ℜ and ℑ intersect.
Students may have difficulty identifying the hypothesis and the conclusion given a conditional statement when then is missing from the statement. Suggest that students first rewrite the statement in the “if, . . . then” form before they identify the hypothesis and conclusion.
Planes ℜ and ℑ intersect in exactly one line. 11. Through any three noncollinear points, there is exactly one plane containing them. Points W, X, and Y are noncollinear.
There is exactly one plane containing points W, X, and Y. 12. If the sum of the digits of an integer is divisible by 3, then the number is divisible by 3. The sum of the digits of 46,125 is 18, which is divisible by 3.
The number 46,125 is divisible by 3.
AVOID COMMON ERRORS
Identify the hypothesis and conclusion of each statement.
Remind students to combine like terms when they solve equations they write by applying the Segment and Angle Addition Postulates.
13. If the ball is red, then it will bounce higher.
Hypothesis: the ball is red
Conclusion: it will bounce higher
14. If a plane contains two lines, then they are coplanar. Hypothesis: the plane contains the two lines Conclusion: the lines are coplanar
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 After students solve for the variable using the
15. If the light does not come on, then the circuit is broken.
Hypothesis: the light does not come on
Conclusion: the circuit is broken
16. You must wear your jacket if it is cold outside.
Hypothesis: it is cold outside
Conclusion: you must wear your jacket
Segment and Angle Addition Postulates, have them use the value to find the actual lengths or angle measures represented by the expressions. They can also use this method to check their work.
Use a definition, postulate, or theorem to find the value of x in the figure described. 17. Point E is between points D and F. If DE = x - 4, EF = 2x + 5, and DF = 4x − 8, find x.
© Houghton Mifflin Harcourt Publishing Company
Use the Segment Addition Postulate; DE + EF = DF; (x − 4) + (2x + 5) = 4x − 8; 3x + 1 = 4x − 8; 9 = x
_ 18. Y is the midpoint of XZ. If XZ = 8x − 2 and YZ = 2x + 1, find x.
―
Because Y is the midpoint of XZ, XY = YZ. Use this fact and the Segment Addition Postulate; XY + YZ = XZ; (2x + 1) + (2x + 1) = 8x − 2; 4x + 2 = 8x − 2; 4 = 4x; 1 = x
→ ‾ is an angle bisector of ∠RST. If m∠RSV = (3x + 5)° and m∠RST = (8x − 14)°, find x. 19. SV → ‾ is an angle bisector of ∠RST, m∠RSV = m∠TSV. Use this fact and Because SV the Angle Addition Postulate; m∠RSV + m∠VST = m∠RST; (3x + 5) + (3x + 5) = 8x - 14; 6x + 10 = 8x - 14; 24 = 2 x; 12 = x 20. ∠ABC and ∠CBD are a linear pair. If m∠ABC = m∠CBD = 3x - 6, find x.
Use the Linear Pair Theorem.; m∠ABC + m∠CDB = 180°; (3x - 6)°+ (3x - 6)° = 180°; 6x - 12 = 180; 6x = 192; x = 32 Module 16
Exercise
IN1_MNLESE389762_U7M16L4 824
Lesson 4
824
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
26
2 Skills/Concepts
MP.3 Logic
27
3 Strategic Thinking
MP.3 Logic
28
2 Skills/Concepts
MP.6 Precision
4/19/14 12:06 PM
Reasoning and Proof 824
Use the figure for Exercises 21 and 22.
PEERTOPEER DISCUSSION
21. Name three collinear points.
Have students use a ruler to draw a line segment made from two non-overlapping segments. Have them label the lengths of one section and the total section. Then have them exchange papers and explain how to use deductive reasoning to find the missing length.
S
Possible answer: P, R, and T
P R
22. Name two linear pairs. Possible answers: ∠PRQ and ∠QRT, ∠STR and ∠UTR
Q
T U
Explain the error in each statement. 23. Two planes can intersect in a single point.
When two planes cross, they intersect each other at an infinite number of points, i.e., in a line.
JOURNAL
24. Three points have to be collinear.
Have students write examples of the following: a conjecture, a counterexample to a conjecture, inductive reasoning, and deductive reasoning.
The three points could be the vertices of a triangle. 25. A line is contained in exactly one plane
A line can be in more than one plane.
26. If x 2 = 25, then x = 5.
The value of x could also be -5.
© Houghton Mifflin Harcourt Publishing Company
H.O.T. Focus on Higher Order Thinking
27. Analyze Relationships What is the greatest number of intersection points 4 coplanar lines can have? What is the greatest number of planes determined by 4 noncollinear points? Draw diagrams to illustrate your answers.
Four coplanar lines can intersect in up to 6 points.
Module 16
IN1_MNLESE389762_U7M16L4 825
825
Lesson 16.4
825
Up to four planes can be determined by 4 noncollinear points.
Lesson 4
4/19/14 12:06 PM
28. Justify Reasoning Prove the Linear Pair Theorem. Given: ∠MJK and ∠MJL are a linear pair of angles. Prove: ∠MJK and ∠MJL are supplementary.
LANGUAGE SUPPORT
M K
Complete the proof by writing the missing reasons. Choose from the following reasons.
J
Angle Addition Postulate
Definition of linear pair
Substitution Property of Equality
Given
Statements
A conjecture is an opinion or proposition that is supported by evidence but has not been proven. It begins with an observation, such as: “I added twenty pairs of odd numbers, and the sums were all even.” The conjecture based on this might be, “The sum of two odd numbers is always even.” A conjecture doesn’t have to be true. For example: “I saw eleven kids with apples at lunch today, and all the apples were red. Conjecture: All apples are red.”
L
Reasons
1. ∠MJK and ∠MJL are a linear pair. → → ‾ are opposite rays. ‾ and JK 2. JL → → ‾ and JK ‾ form a straight line. 3. JL
1. Given
4. m∠LJK = 180°
4. Definition of straight angle
5. m∠MJK + m∠MJL = m∠LJK
5. Angle Addition Postulate
6. m∠MJK + m∠MJL = 180°
6. Substitution Property of Equality
7. ∠MJK and ∠MJL are supplementary.
7. Definition of supplementary angles
2. Definition of linear pair 3. Definition of opposite rays
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to state a possible conjecture for the statement, then support or refute it:
Lesson Performance Task
“I’ve seen thousands of creatures with wings and all of them could fly.” Conjecture: If a creature has wings, then it can fly. Refute: Penguins and ostriches have wings but cannot fly.
If two planes intersect, then they intersect in exactly one line. Find a real-world example that illustrates the postulate above. Then formulate a conjecture by completing the following statement: .
Justify your conjecture with real-world examples or a drawing.
Possible example: a wall and the ceiling Students may make any of the following conjectures: If three planes intersect, then they intersect in a point. If three planes intersect, then they intersect in a line. If three planes intersect, then they intersect in either a point or a line. Obviously the third conjecture is the most complete. Possible examples: two walls and the ceiling intersect at a point; the pages of a book are planes that intersect in a line, the spine of the book.
Module 16
826
“I’ve seen hundreds of bicycles and all of them had two wheels.” Conjecture: If a vehicle is a bicycle, then it has two wheels. Support: The prefix bimeans two, and cycle refers to a wheel. So, the definition of bicycle includes the requirement that the vehicle have two wheels.
© Houghton Mifflin Harcourt Publishing Company
If three planes intersect, then
Lesson 4
EXTENSION ACTIVITY IN1_MNLESE389762_U7M16L4 826
The “impossible” triangle in the photo at the beginning of the lesson is an example of an optical illusion. Have students research optical illusions and either sketch or print ones that especially intrigue them. For each illusion, students should describe what’s intriguing about it, and then explain—if they can!—how the illusion is accomplished.
4/19/14 12:06 PM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Reasoning and Proof 826
MODULE
16
MODULE
STUDY GUIDE REVIEW
Study Guide Review
Essential Question: How can you use tools of geometry to solve real-world problems?
ASSESSMENT AND INTERVENTION
KEY EXAMPLE 6+3 5+1 _ , (_ 2 2 )
( ) ( )
9 6 , __ = __ 2 2 9 = 3, __ 2
Assign or customize module reviews.
Apply the midpoint formula. Simplify the numerators.
ray (rayo) coplanar (coplanares) parallel (paralelo) collinear (colineales) postulate (postulado)
Simplify.
→ ‾ is the angle bisector of ∠ABC so it divides the angle into two BD angles of equal measure. Then m∠ABD + m∠DBC = m∠ABC and m∠ABD = m∠DBC.
COMMON CORE
So, 2 · m∠ABD = m∠ABC. m∠ABD = 20°
Mathematical Practices: MP.1, MP.2, MP.3, MP.4, MP.6 G-CO.A.1
Substitute the angles and simplify.
KEY EXAMPLE
• How to find their current GPS location: The best way is to use a smartphone app or a map search engine. • A good representation of their GPS location: A good representation is latitude and longitude given in decimal degrees, not in degrees, minutes, and seconds. The GPS coordinates are then expressed in (x, y) notation as the ordered pair (latitude, longitude). • What the decimal approximation using the distance formula represents, and what units are used: distance measured in degrees
© Houghton Mifflin Harcourt Publishing Company
SUPPORTING STUDENT REASONING
(Lesson 16.3)
Use the rule (x, y) → (x + 1, 2y) and the points of a triangle, A(1, 2), B(2, 4), and C(2, 2) to draw the image. Determine whether this is a rigid motion. A'(1 + 1, 2(2)), B'(2 + 1, 2(4)),
C'(2 + 1, 2(2))
A'(2, 4), B'(3, 8), C'(3, 4)
___
A'B' = √ (3 - 2) + (8 - 4) 2
2
_
= √ 17 ≈ 4.1
___
AB = √(2 - 1) + (4 - 2) 2
point (punto)
line segment (segmento de línea) endpoints (punto final)
KEY EXAMPLE (Lesson 16.2) → ‾ is the angle bisector of ∠ABC and m∠ABC = 40°. The ray BD Find m∠ABD.
MODULE PERFORMANCE TASK
Key Vocabulary line (línea) plane (plano)
(Lesson 16.1)
Find the midpoint of (5, 6) and (1, 3).
Students should begin this problem by focusing on what information they will need. Here are some issues they might bring up.
16
Tools of Geometry
2
_
= √ 5 ≈ 2.2
Use the transformation rule. Simplify. Use the distance formula to find the distance between A’ and B’. Simplify.
midpoint (punto medio) segment bisector (segmento bisectriz) angle (ángulo) vertex (vértice) side (lado) degrees (grados) angle bisector (bisectriz de un ángulo) transformation (transformación) preimage (preimagen) image (imagen) rigid motion (movimiento rígido) conjecture (conjetura) inductive reasoning (razonamiento inductivo) deductive reasoning (razonamiento deductivo) theorem (teorema) counterexample (contraejemplo) conditional statement (sentencia condicional) linear pair (par lineal)
Use the distance formula to find the distance between A and B. Simplify.
The image is not a rigid motion because the side lengths are not equal. Module 16
827
Study Guide Review
SCAFFOLDING SUPPORT
IN1_MNLESE389762_U7M16MC 827
• Show students how to look up their current GPS location by using a computer search engine or a smartphone app, or give students their current location in decimal degrees. • Students can use the conversion factor 1 degree = 69.2 miles to find distances. The Sample Solution on the next page mentions limitations associated with this value. • Suggest that students make calculations to the nearest thousandth and round to the nearest hundredth.
827
Module 16
• Distance calculations in this task are quite complex. You may wish to have students work on the task in pairs.
4/19/14 12:54 PM
EXERCISES Find the midpoint of the pairs of points. (Lesson 16.1)
(4, 7) and (2, 9)
1.
(3, 8)
2.
SAMPLE SOLUTION
(2, 4)
(5, 5) and (-1, 3)
Current location: 33.57°N, 101.88°W Distance, current location to Austin:
―――――――――――― ―――――― 27.602 ≈ 5.25° √(3.26) + (4.12) ≈ √―――
Find the measure of the angle formed by the angle bisector. (Lesson 16.2) → ‾ is the angle bisector of ∠ABC and m∠ABC = 110°. Find m∠ABD. 3. The ray BD
√(33.57 - 30.31) 2 + (101.88 - 97.76) 2 =
55°
2
Use the rule (x, y) → (3x, 2y) to find the image for the preimage defined by the points. Determine whether the transformation is a rigid motion. (Lesson 16.3)
5.25° × 69.2 miles /° = 363.3 miles
4. A(3, 5), B(5, 3), C(2, 2)
Distance, current location to Columbus:
The points of the image are A'(9, 10), B'(15, 6), C'(6, 4) . is not a rigid motion. The image
―――――――――――― ―――――― 397.92 ≈ 19.95° √(6.41) + (18.89) ≈ √―――
√(39.98 - 33.57) 2 + (101.88 - 82.99) 2 =
5. The child chose Rock in all four games of Rock-Paper-Scissors. The child always chooses Rock.
19.95° × 69.2 miles /° = 1380.54 miles
The conjecture uses inductive reasoning.
Distance, current location to Nashville:
―――――――――――― ―――――― 234.77 ≈ 15.32° √(2.6) + (15.1) ≈ √―――
√(36.17 - 33.57) 2 + (101.88 - 86.78) 2 =
MODULE PERFORMANCE TASK
How Far Is It?
2
Many smartphone apps and online search engines will tell you the distances to nearby restaurants from your current location. How do they do that? Basically, they use latitude and longitude coordinates from GPS to calculate the distances. Let’s explore how that works for some longer distances.
City
Which of the state capitals do you think is nearest to you? Which is farthest away? Use the distance formula to calculate your distance from each of the cities in degrees. Then convert each distance to miles.
Latitude
Module 16
Distance, current location to Sacramento:
Longitude
Austin, TX
30.31° N
97.76° W
Columbus, OH
39.98° N
82.99° W
Nashville, TN
36.17° N
86.78 ° W
Sacramento, CA
38.57° N
121.5° W
Use an app or search engine to find the distance between your location and Your Location each of the capital cities. How do these distances compare with the ones you calculated? How might you account for any differences?
828
2
15.32° × 69.2 miles /° = 1060.14 miles
―――――――――――― ――――― 409.94 ≈ 20.24° √(5) + (19.62) ≈ √―――
√(38.57 - 33.57) 2 + (121.5 - 101.88) 2 =
© Houghton Mifflin Harcourt Publishing Company
The table lists latitude and longitude for four state capitals. Use an app or search engine to find the latitude and longitude for your current location, and record them in the last line of the table.
•
2
2
Determine whether the conjecture uses inductive or deductive reasoning. (Lesson16.4)
•
2
2
2
20.24° × 69.2 miles /° = 1400.61 miles Error, current location to Austin: 363.3 - 333 = 30.3 miles Error, current location to Columbus: 1380.54 -1133 = 247.54 miles Error, current location to Nashville: 1060.14 - 876 = 184.14 miles Error, current location to Sacramento: 1400.61 - 1146 = 254.61 miles
Study Guide Review
DISCUSSION OPPORTUNITIES
IN1_MNLESE389762_U7M16MC 828
• What are other ways to locate places on Earth? • Why are some GPS coordinates expressed with negative numbers? A negative longitude represents a coordinate in the Western hemisphere (a positive longitude is in the Eastern hemisphere), while a negative latitude represents a coordinate in the Southern hemisphere (a positive latitude is in the Northern hemisphere).
4/19/14 12:54 PM
Possible error sources: Calculated distances are straight-line distances, which will be less than distances measured along the curved surface of the Earth; and although 1 degree of latitude measures 69.2 miles, degrees of longitude range from 69.2 miles at the Equator to 0 miles at the North Pole. Precise distance calculations would need to take this into account. Assessment Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate
Study Guide Review 828
Ready to Go On?
Ready to Go On?
16.1–16.4 Tools of Geometry
ASSESS MASTERY Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
• Online Homework • Hints and Help • Extra Practice
Use a definition, postulate, or theorem to find the value desired.
1. Point M is the midpoint between points A(-5, 4) and B(-1, -6). Find the location of M. (Lesson 16.1)
) (
)
Use the midpoint formula. y1 + y2 x1 + x2 _ -5 + (-1) 4 + (-6) M _ = M __________, _ = M (-3, -1) , 2 2 2 2
(
ASSESSMENT AND INTERVENTION
Given triangle EFG, graph its image E'F'G' and confirm that it preserves length and angle measure. (Lesson 16.1) 2. (x, y) → (x - 1, y + 5)
EF = E'F' = 6, FG = F'G' = 8
――――――― GE = √(x - x ) + (y - y ) ―――――――― ( ) ( )
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
=
√
= 10 G'E' =
2
8-2
1
2
2
2
1
+ (-9 - -1
―――――――
)
4
2
-8
√(7 - 1) 2 + (-4 - 4) 2
= 10
m∠F = m∠F' = 90°, m∠E = m∠E' = 53°, and
Response to Intervention Resources
m∠G = m∠G' = 37°
Differentiated Instruction Resources • Reading Strategies • Success for English Learners • Challenge Worksheets Assessment Resources
© Houghton Mifflin Harcourt Publishing Company
ADDITIONAL RESOURCES • Reteach Worksheets
y
8
2
-4
F′
E′
0 E -4 -8
x F
4 G′
G
Find the measure of the angle formed by the angle bisector. (Lesson 16.2) _ 3. The ray GJ is the angle bisector of ∠FGH and ∠FGH = 75°. Find m∠FGJ. 37.5° _ 4. The ray XZ is the angle bisector of ∠WXY and m∠WXY = 155°. Find m∠YXZ. 77.5°
ESSENTIAL QUESTION 5. When is a protractor preferred to a ruler when finding a measurement?
• Leveled Module Quizzes
Answers may vary. Sample: When finding the angle between two lines as opposed to finding the length of a line segment.
Module 16
COMMON CORE
Study Guide Review
829
Common Core Standards
IN1_MNLESE389762_U7M16MC 829
829
Module 16
4/19/14 12:54 PM
Content Standards Mathematical Practices
Lesson
Items
15.1
1
G-CO.A.1, G-GPE.B.4
MP.2
15.1, 15.3
2
G-CO.A.1, G-GPE.B.4
MP.7
15.2
3
G-CO.A.1
MP.2
15.2
4
G-CO.A.1
MP.2
MODULE MODULE 16 MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. For two angles, ∠ABC _ and ∠DBC, m∠ABC = 30° and ∠DBC is its supplement. Ray BE is the angle bisector of ∠ABD. Consider each angle. Does the angle have a measure of 45°? Select Yes or No for A–C. A. ∠DBC
Yes Yes Yes
B. ∠ABE C. ∠DBE
ASSESSMENT AND INTERVENTION
No No No
_
_
2. The line y = √x is transformed into y = √5x . Choose True or False for each statement. A. A dilation can be used to obtain this True False transformation. B. A rotation can be used to obtain this transformation. C. A translation can be used to obtain this transformation.
True
False
True
False
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
ADDITIONAL RESOURCES
3. Triangle ABC is given by the points A(1, 1), B(3, 2), and C(2, 3). Consider each rule of transformation. Does the rule result in an image with points A'(2, 2), B'(6, 3), and C'(4, 4)? Select Yes or No for A–C. A. (x, y) → (x, y + 1) B. (x, y) → (2x, 2y) C. (x, y) → (2x, y + 1)
Yes Yes Yes
AVOID COMMON ERRORS
( ) 17 = (1, ___ 2)
Module 16
COMMON CORE
)
17 2, _ = _ 2 2
• Leveled Module Quizzes: Modified, B
Item 4 Some students may substitute into the midpoint formula incorrectly. They may use x and y from the first ordered pair in the first part of the formula, and use x and y from the second ordered pair for the second part. Encourage students to label their points to avoid confusion; for example: (4, 2), (3, –1) x 1, y 1 x 2, y 2
© Houghton Mifflin Harcourt Publishing Company
) (
Assessment Resources
No No No
4. Find the midpoint of (4, 5) and (–2, 12). Show your work. y1 + y2 4 + (-2) 5 + 12 x 1 + x 2 ______ _ , = _, _ 2 2 2 2
(
16
Study Guide Review
830
Common Core Standards
IN1_MNLESE389762_U7M16MC 830
4/19/14 12:54 PM
Content Standards Mathematical Practices
Lesson
Items
15.2
1
G-CO.D.12
MP.2
15.3
2
G-CO.A.2
MP.7
15.3
3
G-CO.A.2
MP.8
15.1
4
G-GPE.B.4
MP.2
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 830
MODULE
17
Transformations and Symmetry
Transformations and Symmetry
Essential Question: How can you use
ESSENTIAL QUESTION:
transformations to solve real-world problems?
Answer: Rigid transformations can represent situations in which objects slide, turn, or flip.
17 MODULE
LESSON 17.1
Translations LESSON 17.2
Reflections LESSON 17.3
Rotations
This version is for Algebra 1 and Geometry only
PROFESSIONAL DEVELOPMENT VIDEO
LESSON 17.4
Investigating Symmetry
Professional Development Video
Professional Development my.hrw.com
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Helen King/Corbis
Author Juli Dixon models successful teaching practices in an actual high-school classroom.
REAL WORLD VIDEO Check out how transformations can be used to cut patterns out of fabric as efficiently as possible.
MODULE PERFORMANCE TASK PREVIEW
Animating Digital Images In this module, you will use transformations to create a simple animation of a bird in flight. How do computer animators use translations, rotations, and reflections? Let’s find out.
Module 17
DIGITAL TEACHER EDITION IN1_MNLESE389762_U7M17MO 831
Access a full suite of teaching resources when and where you need them: • Access content online or offline • Customize lessons to share with your class • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most
831
Module 17
831
PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests.
4/19/14 11:28 AM
Are YOU Ready?
Are You Ready?
Complete these exercises to review the skills you will need for this module.
ASSESS READINESS
Properties of Reflections Example 1
Use the assessment on this page to determine if students need strategic or intensive intervention for the module’s prerequisite skills.
• Online Homework • Hints and Help • Extra Practice
A figure in the first quadrant is reflected over the x-axis. What quadrant is the image in? The image is in the fourth quadrant. A figure drawn on tracing paper can be reflected across the x-axis by folding the paper along the axis.
Find the quadrant of each image. 1.
The image from reflecting a figure in the first quadrant over the y-axis
2.
The image from reflecting a figure in the second quadrant over the x-axis
ASSESSMENT AND INTERVENTION
The second quadrant The third quadrant
Properties of Rotations Example 2
A figure in the first quadrant is rotated 90° counterclockwise around the origin. What quadrant is the image in? The image is in the second quadrant. In the second quadrant, each point of the figure forms a clockwise 90° angle around the origin with its corresponding point in the original figure.
3 2 1
Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student’s individual needs!
Find the quadrant of each image. 3.
The image from rotating a figure in the third quadrant 180° clockwise
The first quadrant
4.
The image from rotating a figure in the first quadrant 360° clockwise
The first quadrant
TIER 1, TIER 2, TIER 3 SKILLS
Properties of Translations A figure in the first quadrant is translated up 3 units and to the right 1 unit. What quadrant is the image in? The image is in the first quadrant. A translation only moves the image in a direction; the image is not reflected or rotated. Answer each question. 5.
6.
A figure in the first quadrant is translated down and to the right. Is it known what quadrant the image is in? No A figure is translated 3 units up and 2 units left. How large is the image in comparison to the figure? They are the same size.
Module 17
IN1_MNLESE389762_U7M17MO 832
Tier 1 Lesson Intervention Worksheets Reteach 17.1 Reteach 17.2 Reteach 17.3 Reteach 17.4
© Houghton Mifflin Harcourt Publishing Company
Example 3
ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: • Tier 2 Skill Pre-Tests for each Module • Tier 2 Skill Post-Tests for each skill
832
Response to Intervention Tier 2 Strategic Intervention Skills Intervention Worksheets 42 Properties of Reflections 43 Properties of Rotations 44 Properties of Translations
Differentiated Instruction
4/19/14 11:28 AM
Tier 3 Intensive Intervention Worksheets available online Building Block Skills 46, 53, 56, 102, 103
Challenge worksheets Extend the Math Lesson Activities in TE
Module 17
832
LESSON
17.1
Name
Translations
Class
Date
17.1 Translations Essential Question: How do you draw the image of a figure under a translation?
Common Core Math Standards The student is expected to: COMMON CORE
Resource Locker
G-CO.A.4
A translation slides all points of a figure the same distance in the same direction. You can use tracing paper to model translating a triangle.
Mathematical Practices COMMON CORE
Exploring Translations
Explore
Develop definitions of ... translations in terms of ... parallel lines, and line segments. Also G-CO.A.1, G-CO.A.2, G-CO.A.5, G-CO.B.6
A
MP.7 Using Structure
Language Objective Work with a partner to identify examples and non-examples of translations.
B
First, draw a triangle on lined paper. Label the vertices A, B, and C. Then draw a line segment XY. An example of what your drawing may look like is shown.
Use tracing paper to draw a copy of triangle _ ABC. Then copy XY so that the point X is on top of point A. Label the point made from Y as A'.
B
ENGAGE
A A
C
Essential question: How do you draw the image of a figure under a translation?
PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo; ask students to describe repetitions they see in the pattern and tell whether they could create it by using copies of a smaller section. Then preview the Lesson Performance Task.
A
C
C
Y
X
© Houghton Mifflin Harcourt Publishing Company
Possible answer: You can use tracing paper to slide the figure parallel to a vector. On the coordinate plane, you can draw the image by using coordinate notation to calculate the coordinates of the images of the vertices.
B
D
Using the same piece of tracing paper, place A' on A and draw a copy of △ABC. Label the corresponding vertices B' and C'. An example of what your drawing may look like is shown.
Use a ruler to draw line segments from each vertex of the preimage to the corresponding vertex on the new image.
B
B
B
B A
A A
Module 17
C'
C
X
Y
X
A
Y
X
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
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Quest Essential COMMON CORE
IN1_MNLESE389762_U7M17L1 833
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Watch for the hardcover student edition page numbers for this lesson.
ABC. point made Label the of point A.
B A B
C
A C
A
Y X Y
X
Publishin
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C' C
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A
Y X Y
X Lesson 1 833 Module 17
7L1 833 62_U7M1
ESE3897
IN1_MNL
833
Lesson 17.1
4/19/14
10:25 AM
4/19/14 10:27 AM
E
Measure the distances AA', BB', CC', and XY. Describe how AA', BB', and CC' compare to the length XY.
EXPLORE
Possible answer: BB', AA', and CC' are each 2 inches. XY is also 2 inches. The segments are
Exploring Translations
all the same length. Reflect
1.
INTEGRATE TECHNOLOGY
Are BB', AA', and CC' parallel, perpendicular, or neither? Describe how you can check that your answer is reasonable. They are parallel. I can turn the tracing paper so that one of the lines is on one of the
If you use software instead of tracing paper, begin by reviewing how to use geometry software to draw points and figures, to mark a vector, and to translate using a vector. Familiarity with those techniques will allow students to concentrate on the concepts they are exploring.
parallel rules of the lined paper. All of the segments line up with the lines on the paper. So, they are parallel. 2.
How does the angle BAC relate to the angle B'A'C' ? Explain. Possible answer: The angles are congruent. Since translation is a rigid transformation, the
side lengths and angle measures remain the same in the translated figure.
Explain 1
Translating Figures Using Vectors
A vector is a quantity that has both direction and magnitude. The initial point of a vector is the starting point. The terminal point of a vector is the ending point. The vector ⇀ shown may be named ‾ EF or ⇀ v.
QUESTIONING STRATEGIES
F ⇀ ν
E
Initial point
How does the given vector relate to the translation of the triangle? It gives both the distance and the direction between pairs of corresponding points from the preimage and the image.
Terminal point
Translation It is convenient to describe translations using vectors. A translation is a transformation along a vector such that the segment joining a point and its image has the same length as the vector and is parallel to the vector.
C′ B
C
EXPLAIN 1
© Houghton Mifflin Harcourt Publishing Company
A′ A B′
⇀ ν For example, BB' is a line segment that is the same length as and is parallel to vector ⇀ v. You can use these facts about parallel lines to draw translations.
• Parallel lines are always the same distance apart and never intersect.
Translating Figures Using Vectors AVOID COMMON ERRORS Some students may assume that they should slide the figure along the vector. Compare a vector to aspects of a map such as scale or location of north and south, since the vector gives information about the direction and distance of the move.
• Parallel lines have the same slope.
Module 17
834
Lesson 1
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M17L1 834
Integrate Mathematical Practices
4/19/14 10:27 AM
This lesson provides an opportunity to address Mathematical Practice MP.7, which calls for students to “look for and make use of structure.” Students are already familiar with translating a figure in the plane; in this lesson, they explore translations using tracing paper, and then describe translations using vectors, both in the plane and in the coordinate plane. Students use vector notation to describe the translation vector in component form, and then relate the vector to the associated algebraic rule used to transform the preimage figure in the coordinate plane.
Translations
834
Draw the image of △ABC after a translation along ⇀ v.
Example 1
COOPERATIVE LEARNING
Have students pair up to check each other’s drawings of the vectors and translated figures. Remind students to check the labels on their partners’ drawings.
A B
⇀ ν
C
Draw a copy of ⇀ v with its initial point at vertex A of △ABC. The copy must be the same length as ⇀ v , and it must be parallel to ⇀ v . Repeat this process at vertices B and C.
Draw segments to connect the terminal points of the vectors. Label the points A', B', and C'. △A'B'C' is the image of △ABC. A′
A
A
B′
B
B C′ ⇀ ν
C
A′
B′
A
D
⇀ ν
Draw a vector from the vertex A that is the same length as _ and parallel to vector v. The terminal point A' will left 5 be units up and 3 units .
C′
D′ ⇀ ν © Houghton Mifflin Harcourt Publishing Company
C
Draw three more vectors that are parallel from, and D with terminal points B', C', and D'.
B
C
B
,
C
,
Draw segments connecting A', B', C', and D' to form quadrilateral A'B'C'D' .
Reflect
3.
How is drawing an image of quadrilateral ABCD like drawing an image of △ABC? How is it different? Possible answer: The steps to drawing an image of the quadrilateral are the same as
drawing an image of the triangle. It is different because there is an extra vector when drawing the quadrilateral. Also, you must be careful to connect the vertices in the same order as the original shape.
Module 17
835
Lesson 1
COLLABORATIVE LEARNING IN1_MNLESE389762_U7M17L1 835
Peer-to-Peer Activity Have students work in pairs. Instruct one student in each pair to give an example of a translation that would move a given figure from Quadrant I to Quadrant III, and write steps to show what was done. The partner then checks this work. Then they switch roles and repeat, using a translation between two different quadrants.
835
Lesson 17.1
4/19/14 10:27 AM
A
Your Turn
Draw the image of △ABC after a translation along ⇀ v.
4.
B
EXPLAIN 2
A′ B′
Drawing Translations on a Coordinate Plane
C C′
Explain 2
⇀ ν
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Relate the concept of slope, the ratio of the
Drawing Translations on a Coordinate Plane
A vector can also be named using component form, 〈a, b〉, which specifies the horizontal change a and the vertical change _b from the initial point to the terminal point. The component form for PQ is 〈5, 3〉.
Q 3
You can use the component form of the vector to draw coordinates for a new image on a coordinate plane. By using this vector to move a shape, you are moving the x-coordinate 5 units to the right. So, the new x-coordinate would be 5 greater than the x-coordinate in the preimage. Using this vector you are also moving the y-coordinate up 3 units. So, the new y-coordinate would be 3 greater than the y-coordinate in the preimage.
P
vertical change to the horizontal change between two points, to the component form of the vector.
5
QUESTIONING STRATEGIES
Rules for Translations on a Coordinate Plane Translation a units to the right
(x, y) → (x + a, y)
What is the slope of the line that contains the vector (a, b )? ba
_
(x, y) → (x – a, y)
Translation a units to the left
(x, y) → (x, y + b)
Translation b units up
Describe how two vectors could have the same slope but not be the same vector. They could have different magnitudes.
(x, y) → (x, y – b)
Translation b units down
So, when you move an image to the right a units and up b units, you use the rule (x, y) → (x + a, y + b) which is the same as moving the image along vector 〈a, b〉.
Calculate the vertices of the image figure. Graph the preimage and the image.
Preimage coordinates: (−2, 1), (−3, -2), and (−1, −2). Vector: 〈4, 6〉 Predict which quadrant the new image will be drawn in: 1 st quadrant.
Then use the preimage coordinates to draw the preimage, and use the image coordinates to draw the new image.
Use a table to record the new coordinates. Use vector components to write the transformation rule.
Preimage coordinates (x, y)
y
4
(–2, 1)
(–3, –2)
(–1, –2)
Module 17
A
6
Image (x + 4, y + 6)
A′
(2, 7)
-2
(1, 4)
C′
(3, 4) 836
2 0
C
B ⇀ ν 2
AVOID COMMON ERRORS
© Houghton Mifflin Harcourt Publishing Company
Example 2
Some students may try to position the preimage at the beginning of the vector rather than the reverse. Help them position the vector at the vertices of the preimage.
AVOID COMMON ERRORS Students sometimes confuse translation with transformation. Have them draw a Venn diagram to make clear that translation, or slide, is one kind of transformation.
x 4
B′
Lesson 1
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U7M17L1 836
Critical Thinking
Remind students that they use the absolute value of the difference of coordinates when finding a vertical or horizontal distance. Ask, why don’t we use the absolute value when finding the component form of a vector? Direction is an important part of a vector, whether positive or negative. Using the absolute value would make all components positive.
CONNECT VOCABULARY 4/19/14 10:27 AM
Relate the word translation in geometry to translating languages. For example, when we translate a sentence from English to Spanish, we want to keep the same meaning, from each word, for the translation to be accurate. When we translate an image, every point and line in that image is moved in the same direction and the same distance.
Translations
836
Preimage coordinates: A(3, 0), B(2, −2), and C(4, −2). Vector 〈−2, 3〉
EXPLAIN 3
Some students may make an error when they subtract negative coordinates. Review the rules for subtracting integers, as needed.
Image
(2, −2) (4, −2)
(1 (0 (2
, , ,
) 1 ) 1 )
)
y 4
A′
2
3
-2
C′
B′ 0
x
A 2
-2
B
4 C
Your Turn
QUESTIONING STRATEGIES
Draw the preimage and image of each triangle under a translation along 〈−4, 1〉.
In writing the component form of the vector, how would you indicate that there is no vertical change in the position of the figure? Use the number 0 to indicate no change.
5.
Triangle with coordinates: A(2, 4), B(1, 2), C(4, 2). 4
INTEGRATE MATHEMATICAL PRACTICES Cognitive Strategies © Houghton Mifflin Harcourt Publishing Company
Triangle with coordinates: P(2, –1), Q(2, –3), R(4, –3). y
A
C′
2 -2
6.
y
A′
B′
Ask students which method for specifying the translation of a figure they find most helpful in visualizing the image. Have them discuss the advantages of each kind of description.
.
x− 2 ,y+ 3
(3, 0)
AVOID COMMON ERRORS
INTEGRATE MATHEMATICAL PRACTICES Multiple Representations
(
Preimage coordinates (x, y)
Specifying Translation Vectors
Discuss what happens when a point moves from one end of a vector to the other end, and how this relates to the translation of the vertices in the image figure.
1
Prediction: The image will be in Quadrant
B
2
C
0
2
P′ -2
x
Q′
4
x 0 -2
2P R′
-2
Q
4 R
Specifying Translation Vectors
Explain 3
You may be asked to specify a translation that carries a given figure onto another figure. You can do this by drawing the translation vector and then writing it in component form. Example 3
Specify the component form of the vector that maps △ABC to △A'B'C'.
A
4 C
Determine the components of ⇀ v.
y
2 B -4
⇀ ν
0
x
A′
-2
2
4 C′
-4 B′
Module 17
The horizontal change from the initial point (−4, 1) to the terminal point (1, −3) is 1 − (−4) = 5. The vertical change from the initial point (−4, 1) to the terminal point (1, −3) is −3 − 1 = −4 Write the vector in component form. ⇀ v = 〈5, -4〉
837
Lesson 1
LANGUAGE SUPPORT IN1_MNLESE389762_U7M17L1 837
Connect Vocabulary Help students understand the difference between the initial, or beginning, point of a vector and the terminal, or ending, point. Discuss the effects of reversing the points. When talking about the component form of a vector, emphasize the importance of listing the horizontal (left and right) change first, and then the vertical (up and down) change. Point out how reversing the two numbers changes the definition of the vector and, therefore, the location of the image it defines.
837
Lesson 17.1
4/19/14 10:27 AM
Draw the vector ⇀ v from a vertex of △ABC to its image in △A'B'C'.
y
B 4 ⇀ ν
A C
-2
2 B 0
-2
A′
B′
C′ 2
ELABORATE
Determine the components of ⇀ v. x 4
The horizontal change from the initial point (–3, 1) to the terminal point (2, 4) is 2 – -3 = 5 .
QUESTIONING STRATEGIES What can you say about the length of the line segments drawn from each vertex to its image? Each is equal to the magnitude of the vector used to draw the translation.
The vertical change from the initial point to the terminal point is 4 – 1 = 3 Write the vector in component form.
⇀ v =
⟨
5 , 3
⟩
Reflect
7.
What is the component form of a vector that translates figures horizontally? Explain. The vector has the form ⟨a, 0⟩. There is no change in the vertical direction, so the value of
SUMMARIZE THE LESSON Give at least three different ways to describe a translation. Sample answer: A translation is (1) a slide that moves the figure so many units horizontally and so many vertically; (2) moving all points of the figure the same distance in the same direction; (3) drawing an image by sliding a preimage parallel to a given vector a distance equal to the length of the vector; (4) using a pair of coordinates to find the coordinates of the images of the vertices.
b is 0. Your Turn
8.
In Example 3A, suppose △A'B'C' is the preimage and △ABC is the image after translation. What is the component form of the translation vector in this case? How is this vector related to the vector you wrote in Example 3A? 〈-5, 4〉; the components are the opposites of the components of the vector in Example 3A.
Elaborate 9.
How are translations along the vectors 〈a, −b〉 and 〈−a, b〉 similar and how are they different? They move the points the same distance, but in opposite directions.
11. A translation along the vector 〈a, b〉 maps points in Quadrant I to points in Quadrant III. What can you conclude about a and b? Justify your response. Both a and b are negative. Points in Quadrant I have positive x- and y-coordinates and
points in Quadrant III have negative x- and y-coordinates. This means the horizontal and vertical changes are both negative. So a is negative and b is negative.
© Houghton Mifflin Harcourt Publishing Company
10. A translation along the vector 〈−2, 7〉 maps point P to point Q. The coordinates of point Q are (4, −1). What are the coordinates of point P? Explain your reasoning. (6, −8); Solving equations x − 2 = 4 and y + 7 = −1 shows that x = 6 and y = −8.
12. Essential Question Check-In How does translating a figure using the formal definition of a translation compare to the previous method of translating a figure? Possible answer: Rather than sliding a figure left or right and then up or down to translate
it, you slide it parallel to a given vector a distance equal to the length of the vector. Module 17
IN1_MNLESE389762_U7M17L1 838
838
Lesson 1
4/19/14 10:27 AM
Translations
838
Evaluate: Homework and Practice
EVALUATE
Draw the image of △ABC after a translation along ⇀ v. 1.
2.
A
⇀ ν A′
C
Concepts and Skills
Practice
Explore Exploring Translations
Exercise 4
Example 1 Translating Figures Using Vectors
Exercises 3, 8-10
Example 2 Drawing Translations on a Coordinate Plane
Exercises 5–7
Example 3 Specifying Translation Vectors
Exercises 12–14
4.
C′
B′
translated images (preimage and image) and preimages that may have been transformed but are not translations. Have students sort the pictures into translations and non-translations, discuss why they are or are not translations, and write the justifications on note cards. Provide a list of key terms for students to use in their explanations.
Triangle: A(-3, -1); B(-2, 2); C(0, -1); Vector: ⟨3, 2⟩
© Houghton Mifflin Harcourt Publishing Company
A
⇀ ν
B
C A′ A
-2
Triangle: P(1, -3); Q(3, -1); R(4, -3); Vector: ⟨-1, 3⟩
y B′
2 A′ 0
6.
2 C′
C
2
-2
x
4
R′ 0
C′
B′
B
Y
Triangle: X(0, 3); Y(−1, 1); Z(–3, 4); Vector: ⟨4, -2⟩ Z
2 P
y X
2 -2
R
9.
4
x
Q′ Q4
-2
Y
Name the vector. Write it in component form.
Z′ X′
0
–––⇀, ⟨5, -2⟩ GH
(x, y) → (x + 6, y - 11) (2, -3) → (8, -14)
(3, 1) → (9, -10)
7.
y P′
Find the coordinates of the image under the transformation ⟨6, -11⟩.
8.
2
Y′
x
4
G H
(4, -3) → (10, -14)
10. Match each set of coordinates for a preimage with the coordinates of its image after applying the vector ⟨3, -8⟩. Indicate a match by writing a letter for a preimage on the line in front of the corresponding image. C A. (1, 1); (10, 1); (6, 5) (6, -10); (6, -4); (9, -3) B. (0, 0); (3, 8); (4, 0); (7, 8)
D
(1, -6); (5, -6); (-1, -8); (7, -8)
C. (3, -2); (3, 4); (6, 5)
A
(4, -7); (13, -7); (9, -3)
D. (-2, 2); (2, 2); (-4, 0); (4, 0)
B
(3, -8); (6, 0); (7, -8); (10, 0)
Module 17
IN1_MNLESE389762_U7M17L1 839
Exercise
Lesson 1
839
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
2 Skills/Concepts
MP.4 Modeling
1 Recall of Information
MP.2 Reasoning
2 Skills/Concepts
MP.4 Modeling
8
1 Recall of Information
MP.6 Precision
9
1 Recall of Information
MP.2 Reasoning
10
2 Skills/Concepts
MP.5 Using Tools
1–3
5–7
Lesson 17.1
C
Draw the preimage and image of each triangle under the given translation.
4
839
B′
X
B
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Provide students with graphic examples of
C′
_ _ Line segment XY was used to draw a copy of △ABC. XY is 3.5 centimeters long. What is the length of AA' + BB' + CC'? 10.5 cm
4
constructing the translated image, where the image will lie, using the distance and direction given by the translation vector.
A
B
C
5.
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Encourage students to predict, before
A′
B′
C′
⇀ ν
ASSIGNMENT GUIDE
A′
A
B
• Online Homework • Hints and Help • Extra Practice
3.
4/19/14 10:27 AM
11. Persevere in Problem Solving Emma and Tony are playing a game. Each draws a triangle on a coordinate grid. For each turn, Emma chooses either the horizontal or vertical value for a vector in component form. Tony chooses the other value, alternating each turn. They each have to draw a new image of their triangle using the vector with the components they chose and using the image from the prior turn as the preimage. Whoever has drawn an image in each of the four quadrants first wins the game.
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Have students draw a vector on graph paper
Emma’s initial triangle has the coordinates (-3, 0), (-4, -2), (-2, -2) and Tony’s initial triangle has the coordinates (2, 4), (2, 2), (4, 3). On the first turn the vector ⟨6, -5⟩ is used and on the second turn the vector ⟨-10, 8⟩ is used. What quadrant does Emma need to translate her triangle to in order to win? What quadrant does Tony need to translate his triangle to in order to win?
and then use it as the hypotenuse of a right triangle. Discuss how the legs of the triangle show the vertical and horizontal components.
Emma: Quadrant I; Tony: Quadrant III Specify the component form of the vector that maps each figure to its image. 12.
13. 4
B
4
y
A
B
A 0
C
-2
C′
D
x
-4
4
C -2 A′
0
D′
⟨2, 3⟩
C
x -4
2 B′
A
B A′
C
C′
B -2
0
When writing the component form of a given vector, some students may reverse the initial point and the terminal point. Suggest that they circle the arrow that shows the terminal point before writing the components.
B′ x
2
4
-2 -4
C′
⟨2, -5⟩
15. Explain the Error Andrew is using vector ⇀ v to draw a copy of △ABC. Explain his error.
y A′
4
2
2 -2
A
B′
A′
AVOID COMMON ERRORS
14.
y
⟨5, 0⟩
Possible answer: He drew vectors from A to A' and from B to B' that were not parallel to or the same v . The correct vectors length as ⇀ should each point 3 units right and 3 units down.
⇀ ν
© Houghton Mifflin Harcourt Publishing Company
B′ C′
16. Explain the Error Marcus was asked to identify the vector that maps △DEF to △D'E'F'. He drew a vector as shown and determined that the component form of the vector is ⟨3, 1〉. Explain his error.
4
D
y D′
2
He should have drawn the vector from F to F’ or from E to E’. The correct component form of the vector is ⟨5, 1〉.
-4
x
0
-2 E
F
E′
F′
-4
Module 17
Exercise
IN1_MNLESE389762_U7M17L1 840
Lesson 1
840
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
3 Strategic Thinking
MP.5 Using Tools
12–14
2 Skills/Concepts
MP.4 Modeling
15–16
3 Strategic Thinking
MP.3 Logic
17
2 Skills/Concepts
MP.3 Logic
18
3 Strategic Thinking
MP.3 Logic
2 Skills/Concepts
MP.3 Logic
3 Strategic Thinking
MP.3 Logic
11
19–20 21
4/19/14 10:26 AM
Translations
840
17. Algebra A cartographer is making a city map. Line m represents Murphy Street. The cartographer translates points on line m along the vector ⟨2, -2〉 to draw Nolan Street. Draw the line for Nolan Street on the coordinate plane and write its equation. What is the image of the point (0, 3) in this situation?
JOURNAL Have students describe what a vector is and how it is used to define a translation.
y = 2x - 3; (2, 1)
y x -4 m
0
2
4
-2
H.O.T. Focus on Higher Order Thinking
18. Represent Real-World Problems A builder is trying to level out some ground with a front-end loader. He picks up some excess dirt at (9, 16) and then maneuvers through the job site along the vectors ⟨-6, 0⟩, ⟨2, 5⟩, ⟨8, 10⟩ to get to the spot to unload the dirt. Find the coordinates of the unloading point. Find a single vector from the loading point to the unloading point.
(13, 31); ⟨4, 15⟩
19. Look for a Pattern A checker player’s piece begins at K and, through a series of moves, lands on L. What translation vector represents the path from K to L?
K
⟨4, -6⟩
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t)©Hero Images/Getty Images; (b)©Igor Golovnov/Alamy
L
20. Represent Real-World Problems A group of hikers walks 2 miles east and then 1 mile north. After taking a break, they then hike 4 miles east to their final destination. What vector describes their hike from their starting position to their final destination? Let 1 unit represent 1 mile.
⟨6, 1⟩
21. Communicate Mathematical Ideas In a quilt pattern, a polygon with vertices (-4, -2), (-3, -1), (-2, -2), and (-3, -3) is translated repeatedly along the vector ⟨2, 2⟩. What are the coordinates of the third polygon in the pattern? Explain how you solved the problem.
(0, 2); (1, 3); (2, 2); (1, 1); Possible answer:
I used a table to find the coordinates of the second polygon. Then I made a new table, using the coordinates from the second polygon to find the coordinates of the third polygon.
Module 17
IN1_MNLESE389762_U7M17L1 841
841
Lesson 17.1
841
Lesson 1
4/19/14 10:26 AM
Lesson Performance Task
CONNECT VOCABULARY Ask students to describe any similarities between the meaning of translation as it is used in this lesson and the word’s meaning when it is used to describe the process of converting words from one language to another. Sample answer: Each transforms something into something else. In this lesson, a translation transforms a point to a point in another position. A language translation transforms a word in one language into an equivalent word in another language.
A contractor is designing a pattern for tiles in an entryway, using a sun design called Image A for the center of the space. The contractor wants to duplicate this design three times, labeled Image B, Image C, and Image D, above Image A so that they do not overlap. Identify the three vectors, labeled ⇀ m, ⇀ n , and ⇀ p that could be used to draw the design, and write them in component form. Draw the images on grid paper using the vectors you wrote.
Image A
QUESTIONING STRATEGIES
Drawings and vectors will vary. Possible drawing and vectors are shown.
→= (3, –5) translates P(4, –2) to Q. Suppose m ‾ How do you find the coordinates of Q? Add 3 to 4 for the x-coordinate and add –5 to –2 for the y-coordinate.
Image C Image B
Image D
What is a vector → ‾ n that will translate Q to P? Why? → n = –3, 5 ( ); each component is the ‾ opposite of the component in the translation vector.
Image A © Houghton Mifflin Harcourt Publishing Company
⇀ m moves Image A to Image B. ⇀ m = ⟨-8, 6⟩ ⇀ n moves Image A to Image C. ⇀ n = ⟨0, 8⟩ ⇀ p moves Image A to Image D. ⇀ p = ⟨8, 6⟩
Module 17
842
Lesson 1
EXTENSION ACTIVITY IN1_MNLESE389762_U7M17L1 842
Provide students with grid paper. Have each student design and color a geometric shape. The design should be no larger than a 6 × 6 portion of the grid. The student should write three or more vectors indicating where additional designs like theirs are to be drawn, making sure that translated designs do not overlap the original or each other. Students then exchange grids with a partner and draw the translated designs indicated by their partner’s vectors.
4/19/14 10:26 AM
Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Translations
842
LESSON
17.2
Name
Reflections
Class
Date
17.2 Reflections Essential Question: How do you draw the image of a figure under a reflection?
Common Core Math Standards The student is expected to: COMMON CORE
Resource Locker
G-CO.A.4
Explore
Exploring Reflections
Develop definitions of ... reflections ... in terms of ... perpendicular lines … . Also G-CO.A.1, G-CO.A.2, G-CO.A.5, G-CO.B.6, G-CO.D.12, G-MG.A.3
Use tracing paper to explore reflections.
Mathematical Practices
COMMON CORE
MP.5 Using Tools
Draw and label a line ℓ on tracing paper. Then draw and label a quadrilateral ABCD with vertex C on line ℓ.
Language Objective
D
ℓ
B
A
C
Work with a partner to discuss how to determine if a transformation is a reflection.
ENGAGE
Fold the tracing paper along line ℓ. Trace the quadrilateral. Then unfold the paper and draw the image of the quadrilateral. Label it A′ B′ C′ D′.
Essential Question: How do you draw the image of a figure under a reflection?
C
Use a ruler to measure each segment and the two shorter segments formed by its intersection with line ℓ. What do you notice?
Line ℓ bisects each segment. Reflect
In this activity, the fold line (line ℓ ) is the line of reflection. What happens when a point is located on the line of reflection? The image of the point is also on the line of reflection, in the same location as the
preimage. Module 17
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Date
Name
s
ction 17.2 Refle
tion? under a reflec of a figure the image lines … . Also do you draw ion: How perpendicular terms of … ions … in A.3 of … reflect .12, G-MG. p definitions .6, G-CO.D G-CO.A.4 Develo .2, G-CO.A.5, G-CO.B G-CO.A G-CO.A.1, Reflections
COMMON CORE
Exploring
Explore
IN1_MNLESE389762_U7M17L2.indd 843
Use
reflections. to explore tracing paper
ilatera the quadr line ℓ. Trace of the g paper along the image Fold the tracin paper and draw the ′. Then unfold A′ B′ C′ D l. Label it quadrilatera
A
D
Watch for the hardcover student edition page numbers for this lesson.
C
l. B
ℓ
A
C D
Use a its image. ? l ABCD with do you notice of quadrilatera line ℓ. What ct each vertex by each segment and nts to conne formed Draw segme re the angle ℓ. to measu with line protractor a right angle its ent forms formed by nts segm segme Each r two shorte nt and the re each segme notice? measu to you Use a ruler ℓ. What do with line intersection ent. ts each segm the Line ℓ bisec located on a point is ns when What happe of reflection. ion as the Reflect ℓ ) is the line same locat fold line (line tion, in the activity, the of reflec 1. In this ion? on the line reflect also is line of Lesson 2 e of the point The imag
© Houghto
n Mifflin
Harcour t
Publishin
preimage. Module 17
7L2 843
ESE3897
IN1_MNL
Lesson 17.2
ℓ
B
draw and
g paper. Then ℓ. ℓ on tracin C on line label a line with vertex Draw and ilateral ABCD label a quadr
HARDCOVER PAGES 843856
Resource Locker
Quest Essential
62_U7M1
843
Lesson 2
843
gh "File info"
made throu
Class
y g Compan
View the online Engage. Discuss with students what a mirror does and how it reflects a person—the preimage. Ask students to describe similarities and differences between an object and the image of the object in a mirror. Then preview the Lesson Performance Task.
Draw segments to connect each vertex of quadrilateral ABCD with its image. Use a protractor to measure the angle formed by each segment and line ℓ. What do you notice?
Each segment forms a right angle with line ℓ.
1.
PREVIEW: LESSON PERFORMANCE TASK
ℓ
D
© Houghton Mifflin Harcourt Publishing Company
Possible answer: To draw the image of a figure under a refletion across line ℓ, choose a vertex of the figure, vertex A. Draw a segment with an endpoint at vertex A so that the segment is perpendicular to line ℓ and is bisected by line ℓ. Label the other endpoint of the segment A′. Repeat the process with the other vertices of the figure. Connect the images of the vertices in the same order as the preimage to draw the image of the figure.
B
A
843 4/19/14
10:38 AM
4/19/14 11:36 PM
2.
Discussion A student claims that a figure and its reflected image always lie on opposite sides of the line of reflection. Do you agree? Why or why not? No; this is only true when the figure lies entirely on one side of the line of reflection.
EXPLORE
The statement is not true when the line of reflection passes through one or more
Exploring Reflections
points of the figure.
Explain 1
INTEGRATE TECHNOLOGY
Reflecting Figures Using Graph Paper
Perpendicular lines are lines that intersect at right angles. In the figure, line ℓ is perpendicular to line m. The right angle mark in the figure indicates that the lines are perpendicular.
To carry out the Explore using geometry software, first have students construct a figure similar to the given preimage. Then have them construct a line and mark it as the line of reflection. Finally, have students select the preimage and choose how to reflect it.
ℓ m
The perpendicular bisector of a line segment is a line perpendicular to the segment _ at the segment’s midpoint. In the figure, line n is the perpendicular bisector of AB.
n
QUESTIONING STRATEGIES A
B
A reflection across line ℓ maps a point P to its image P ′.
What is the image of the line of reflection? The line of reflection is its own image.
ℓ
_ • If P is not on line ℓ, then line ℓ is the perpendicular bisector of PP ′.
P′
• If P is on line ℓ, then P = P ′.
EXPLAIN 1
P Q=Q′
© Houghton Mifflin Harcourt Publishing Company
Example 1
Draw the image of △ABC after a reflection across line ℓ.
Step 1 Draw a segment with an endpoint at vertex A so that the segment is perpendicular to line ℓ and is bisected by line ℓ. Label the other endpoint of the segment A′.
ℓ
A′
A
Reflecting Figures Using Graph Paper
B
AVOID COMMON ERRORS Some students might confuse the segments drawn to construct a reflection with the vectors used to draw translations. Point out that the vectors for translations all have equal magnitude, but the segments drawn to reflect a figure vary in length.
C
QUESTIONING STRATEGIES Module 17
844
For the term perpendicular bisector, describe the mark in the figure that results from perpendicular, and the marks that result from bisector.
Lesson 2
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M17L2.indd 844
Integrate Mathematical Practices
4/19/14 11:36 PM
This lesson provides an opportunity to address Mathematical Practice MP.5, which calls for students to “use appropriate tools.” Students are already familiar with reflecting a figure in the plane; in this lesson, students use the tools of tracing paper, ruler, and protractor to explore reflections. Students draw perpendicular bisectors on graph paper to draw reflected images and find midpoints to determine the line of reflection.
Reflections 844
Step 2 Repeat Step 1 at vertices B and C.
ℓ
A′
Step 3 Connect points A′, B′, and C′. △A′B′C′ is the image of △ABC. B′
A
C′
B
C
B
Draw a segment with an endpoint at vertex A so that the segment is perpendicular to line ℓ and is bisected by line ℓ. Label the other endpoint of the segment A′.
B
ℓ C
A
Repeat the process at vertices B and C.
C′
Connect points A′, B′, and C′. △A′B′C ′ is the image of △ABC. Step 1 Draw a segment with an endpoint at vertex A so that the segment is perpendicular to line ℓ and is bisected by line ℓ. Label the other endpoint of the segment A′.
A′
B′
Step 2 Repeat Step 1 at vertices B and C. Step 3 Connect points A', B', and C'. △A'B'C' is the image of △ABC. Reflect
3.
How can you check that you drew the image of the triangle correctly? Possible answer: Since a reflection is a rigid motion, it preserves length. Check that each
side of △ABC has the same length as its image in △A'B'C'.
© Houghton Mifflin Harcourt Publishing Company
4.
_ In Part A, how can you tell that AA' is perpendicular to line ℓ? _ Possible answer: Line ℓ forms a diagonal through corners of grid squares. AA' forms a
diagonal through corners of grid squares in the opposite direction.
Your Turn
Draw the image of △ABC after a reflection across line ℓ. 5.
ℓ
6.
A
A A′
A′ C
B
B
C′
B′
C′ B′
Module 17
C 845
Lesson 2
COLLABORATIVE LEARNING IN1_MNLESE389762_U7M17L2.indd 845
Small Group Activity Have students work in small groups to develop and write a list of what they would look for, or check, when they evaluate whether a classmate’s paper shows a correctly drawn reflection.
845
Lesson 17.2
4/19/14 11:36 PM
Explain 2
Drawing Reflections on a Coordinate Plane
EXPLAIN 2
The table summarizes coordinate notation for reflections on a coordinate plane.
Rules for Reflections on a Coordinate Plane (x, y) → (-x, y)
Reflection across the y-axis Reflection across the line y = x Reflection across the line y = -x Example 2
Drawing Reflections on a Coordinate Plane
(x, y) → (x, -y)
Reflection across the x-axis
(x, y) → (y, x)
(x, y) → (-y, -x)
QUESTIONING STRATEGIES
Reflect the figure with the given vertices across the given line.
M(1, 2), N(1, 4), P(3, 3); y-axis
N′ y N 4
Step 1 Find the coordinates of the vertices of the image.
P′
2 M′
A(x, y) → A′(-x, y).
M(1, 2) →
M′(-1, 2)
P(3, 3) →
P ′(-3, 3)
N(1, 4) →
-4
N′(-1, 4)
-2
P
M
0
Describe in words what happens to the coordinates of a point when the point is reflected across the x-axis. The x-coordinate stays the same and the y-coordinate of the image is the opposite of the y-coordinate of the preimage.
x 2
4
Does this mean that the y-coordinate of an image is always a negative number? Explain. No; it is always the opposite of the preimage, but the opposite of a negative number is a positive number.
Step 2 Graph the preimage. Step 3 Predict the quadrant in which the image will lie. Since △MNP lies in Quadrant I and the triangle is reflected across the y-axis, the image will lie in Quadrant II. Graph the image.
D(2, 0), E(2, 2), F(5, 2), G(5, 1); y = x Step 1 Find the coordinates of the vertices of the image.
y
D(2, 0) → D′
0 ,
,
x 2
E(2, 2) → E′
2 ,
2
F(5, 2) → F′
2 ,
5
1 ,
5
G(5, 1) → G′
) ) ) ) )
4 2 -2
0
F
E′ E
D′ D2
Gx 4
2
Step 2 Graph the preimage.
Step 3 Since DEFG lies in Quadrant 1 and the quadrilateral is reflected across the line y = x, the image will lie in Quadrant
ℓ
© Houghton Mifflin Harcourt Publishing Company
( ( ( ( (
A(x, y) → A′
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students describe what is alike and what
yG′ F′
is different in the preimage and the image for a reflection.
COGNITIVE STRATEGIES
.
Graph the image.
Module 17
846
Lesson 2
DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U7M17L2.indd 846
Visual Cues
4/19/14 11:36 PM
Have students determine if given pre-images have been reflected onto an image. Explain that besides folding paper, or tracing and flipping the image, you can check the distance of each point from whichever axis the reflection occurs across (or whichever axis acts as the line of reflection). Encourage them to state whether or not the image is a reflection of the pre-image across the x- or y-axis, and to check whether each point in the image and preimage is the same distance from the x- or y-axis.
To help students see how a reflection is different from the original image, use a mirror and have students draw an object such as someone facing the class with a pencil behind one ear. Then have the person turn so that the class sees the reflection in the mirror and have them sketch the reflection Discuss how the two sketches differ.
Reflections 846
Reflect
EXPLAIN 3
7.
Specifying Lines of Reflection
How would the image of 4MNP be similar to and different from the one you drew in Part A if the triangle were reflected across the x-axis? The image would have the same size and shape, but it would lie in Quadrant IV instead
of Quadrant II. 8.
QUESTIONING STRATEGIES In order for a line to be a line of reflection, what two things must be true about the line and each segment connecting corresponding points of the preimage and image? The line must pass through the midpoint of each segment, and it must be perpendicular to each segment.
A classmate claims that the rule (x, y) → (-x, y) for reflecting a figure across the y-axis only works if all the vertices are in the first quadrant because the values of x and y must be positive. Explain why this reasoning is not correct. The rule says that the image of a point will have an x-coordinate that is the opposite of
the value of the preimage. So, the point (-1, 2) will have the image (1, 2) when reflected across the y-axis. Your Turn
Reflect the figure with the given vertices across the given line. 9.
10. A(-4, -2), B(-1, -1), C(-1, -4); y = -x
S(3, 4), T(3, 1), U(−2, 1), V(−2, 4); x-axis
AVOID COMMON ERRORS
V
4
y
S
4
2
Some students may think that reflection over a line always puts the image in a different quadrant from the preimage. Help them draw examples of reflecting over a line that is not an axis to see why this is not always true.
U -4 U′ -2
2
-2
B′ -4
T′4
A
V′ -4
(x, y) → (x, -y)
© Houghton Mifflin Harcourt Publishing Company
Example 3
-2
C′ x
0 B -2
2
4
-4 C
S′
S'(3, -4), T'(3, -1), U'(-2, -1), V'(-2, -4)
Explain 3
A′
2
T x 0
y
(x, y) → (-y, -x)
A'(2, 4), B'(1, 1), C'(4, 1)
Specifying Lines of Reflection
Given that △A'B'C' is the image of △ABC under a reflection, draw the line of reflection.
_ _ _ Draw the segments AA′, BB′, and CC′. Find the midpoint of each segment. _ -3 + 5 3 + (-1) The midpoint of AA′ is _, _ = (1, 1). 2 2 _ -2 + 2 0 + (-2) The midpoint of BB′ is _, _ = (0, -1). 2 2 _ + (-5) -5 + 3 -1 _ _ The midpoint of CC′ is = (-1, -3). , 2 2 Plot the midpoints. Draw line ℓ through the midpoints.
( ( (
) )
)
4
A
C
-2
y
ℓ x A′
B 0
2 B′
-4
C′
Line ℓ is the line of reflection. Module 17
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Connect Vocabulary Students who are speakers of Spanish may benefit from learning that many polysyllabic English words that end in –or and –al are shared cognates with Spanish. Although pronounced differently, these words are written the same and are identical in meaning. Some examples are vector, bisector, factor, divisor, numerator, denominator, initial, terminal, vertical, horizontal, and final.
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__ _ Draw AA′, BB′, and CC′. Find the midpoint of each segment.
( ( (
) ) )
_ -3 + -5 3 + -1 = The midpoint of AA′ is __ , __ 2 2 _ 2 + -2 3 + -5 = The midpoint of BB′ is __ , __ 2 2
( -4 ,
).
1
ℓ
( 0 , -1 ). (
4
A
A′
y
EXPLAIN 4
B
2 x -2
)
0
2
4
-2
_ B′ -4 5 + 3 -1 + -5 = 4 -3 . The midpoint of CC′ is __ , , __ 2 2 Plot the midpoints. Draw line ℓ through the midpoints. Line ℓ is the line of reflection.
Applying Reflections
C
QUESTIONING STRATEGIES
C′
For reflected light, or for an object such as a ball bouncing off a wall, what does it mean to say “the angle of reflection equals the angle of incidence”? The angle that the object makes as it hits the reflecting surface is equal to the angle at which the object bounces off that surface.
Reflect
11. How can you use a ruler and protractor to check that line ℓ is_ the line of reflection? _ _ Use the ruler to check that line ℓ bisects AA', BB', and CC'. Use the protractor to check that _ _ _ line ℓ is perpendicular to AA', BB', and CC'. Your Turn
Given that △A'B'C' is the image of △ABC under a reflection, draw the line of reflection.
PEERTOPEER DISCUSSION
12.
Have students discuss and explain familiar events involving angles of reflection, such as playing golf, basketball, and soccer, or using light and mirrors.
A
4
y
2 -2 C
0
-2 C′ -4
A′
C′
0
)
2
B
x
4
6
(
)
_ 4+6 7+ 3 midpoints: AA': ____ , _____ = (5, 5); 2 2 _ 3 + (-1) -1 + 7 ______ _____ BB': , 2 = (1, 3); 2
( (
)
_ -3 + (-1) -1 + (-3) CC': _______ , _________ = (-2, -2) 2 2
Explain 4
A′
2
_ -2 + 4 4 + (-2) midpoints: AA': _____ , ______ = (1, 1); 2 2 _ 1+3 3+ 1 ____ ____ BB': 2 , 2 = (2, 2);
( (
ℓ
4
x 4
( )
A C
6
B′ 2
y
B′
)
)
_ 2+4 6+ 2 CC': ____ , ____ = (3, 4) 2 2
Applying Reflections
Example 4
The figure shows one hole of a miniature golf course. It is not possible to hit the ball in a straight line from the tee T to the hole H. At what point should a player aim in order to make a hole in one?
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D
F
C
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-4
13.
ℓ
B
H T 2
4
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Understand the Problem The problem asks you to locate point X on the wall of the miniature golf hole so that the ball can travel in a straight line from T to X and from X to H.
Make a Plan In order for the ball to travel directly from T to X to H, the angle of the ball’s path as it hits the wall must equal the angle of the ball’s path as it leaves the wall. In the figure, m∠1 must equal m∠2. _ Let H ′ be the reflection of point H across BC. Reflections preserve angle measure, so m∠2 = m∠ 3 . Therefore, m∠1 is equal to m∠2 when m∠1 is equal to m∠3. This occurs when T, X , and H ′ are collinear.
H
C
H′
23 X 1
T
B
Solve
_ Reflect H across BC to locate H ′.
(
)
The coordinates of H ′ are 7 , 4 . _ _ _ Draw TH ′ and locate point X where TH ′ intersects BC.
© Houghton Mifflin Harcourt Publishing Company
The coordinates of point X are
(
)
6 , 3.5 .
The player should aim at this point.
Look Back To check that the answer is reasonable, plot point X using the coordinates you found. Then use a protractor to check that the angle of the ball’s path as it hits the wall at point X is equal to the angle of the ball’s path as it leaves the wall from point X. Reflect
14. Is there another path the ball can take to hit a wall and then travel directly to the hole? Explain. ¯ and locate a point Y on AB ¯ so that the ball Yes; use a similar process to reflect H across AB
travels from T to Y to H.
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Your Turn
15. Cara is playing pool. She wants to use the cue ball C to hit the ball at point A without hitting the ball at point B. To do so, she has to bounce the cue ball off the side rail and into the ball at point A. Find the coordinates of the exact point along the side rail that Cara should aim for.
Side 4 Rail 2
y
ELABORATE A x
-2
0 -2
B
QUESTIONING STRATEGIES
4
C
Describe how you would draw the reflection of a figure drawn on graph paper across the line y = - x? Find the coordinates of each vertex of the preimage and change each using the rule. (x,y) → (-y,-x). Then join the new vertices to draw the image.
-4
Reflect point C across the side rail to locate C'. The coordinates of C' are (−3, −2). Locate _ point X where AC' intersects the side rail. The coordinates of point X are (−1, −1). Cara
should aim for the point (−1, −1) along the side rail.
Elaborate 16. Do any points in the plane have themselves as images under a reflection? Explain. Yes; every point on the line of reflection has itself as its image. This is how the reflection
CONNECT VOCABULARY
image is defined for points that lie on the line of reflection.
Compare and contrast a translation and a reflection by having students write the words and then draw an example of each kind of transformation. Have them write some features of each underneath.
17. If you are given a figure and its image under a reflection, how can you use paper folding to find the line of reflection? Fold the paper so that the figure coincides with its image. Then unfold the paper. The
SUMMARIZE THE LESSON How can you check that a drawing of two figures represents a reflection? (1) The two figures must have the same size and shape. (2) There must be a line of reflection that is the perpendicular bisector of every segment joining the vertices of the preimage to the image (unless a vertex is on the line of reflection).
crease is the line of reflection.
image of the vertex. Plot the images of the vertices. Then connect these points to draw the image of the figure.
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18. Essential Question Check-In How do you draw the image of a figure under a reflection across the x-axis? For each vertex of the figure, use the rule (x, y) → (x, −y) to find the coordinates of the
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Evaluate: Homework and Practice
EVALUATE
• Online Homework • Hints and Help • Extra Practice
Use tracing paper to copy each figure and line ℓ. Then fold the paper to draw and label the image of the figure after a reflection across line ℓ. 1.
2.
B C C′
A
ASSIGNMENT GUIDE
A′
ℓ
ℓ
Practice
Explore Exploring Reflections
Exercises 1–4
Example 1 Reflection Figures Using Graph Paper
Exercises 5–8
Example 2 Drawing Reflections on a Coordinate Plane
Exercises 9–12
Example 3 Specifying Lines of Reflection
Exercises 13–16
Example 4 Applying Reflections
Exercises 17–18
A
A′
B′
D
4.
ℓ
ℓ
B
B C
B′ C′
A′
C
A
D
C′
A
A′ D′
image can have any points in common (can touch or overlap). Have students draw examples to explain their conclusions.
Draw the image of △ABC after a reflection across line ℓ. 5.
© Houghton Mifflin Harcourt Publishing Company
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Discuss whether a preimage and its reflected
ℓ
6.
B
ℓ
A
A
A′ C
B B′
A′
C
B′
C′ C′
7. A
B′
8.
ℓ
A
ℓ
B
A′ A′
C
B′ C′
Exercise
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C
B
Module 17
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C
B′
3.
C′
B′
B
Concepts and Skills
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
1–4
1 Recall of Information
MP.5 Using Tools
5–8
1 Recall of Information
MP.6 Precision
9–12
1 Recall of Information
MP.4 Modeling
13–16
2 Skills/Concepts
MP.2 Reasoning
17–18
2 Skills/Concepts
MP.1 Problem Solving
19–20
2 Skills/Concepts
MP.4 Modeling
21
2 Skills/Concepts
MP.4 Modeling
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Reflect the figure with the given vertices across the given line.
y
4
P
Q
2
S
x
R -2 R′ 0
-4 S′
2
P'(−2, −3), Q'(4, −3), R'(−1, 0), S'(−4, −1)
4 B′
A
11. J(−1, 2), K(2, 4), L(4, −1); y = −x y K J'(−2, 1), 4 K'(−4, −2), L'(1, −4) J 2 J′ x -4 -2 0 2 4L
0
2
-2
C4 A′
-4
12. D(−1, 1), E(3, 2), F(4, −1), G(−1, −3); y = x F′ 4 y D -4 -2 G′
-2
K′
B x
-4C′ -2
Q′
P′ -4
Some students may think that any line through the midpoint of a segment joining two vertices (such as AA' ) is the line of reflection. Have them draw a figure and its reflection as well as the segment joining two corresponding vertices. Then have them draw several different lines through the midpoint in order to identify as the line of reflection the only line that is perpendicular to the segment.
A'(3, −3), B'(−1, 3), C'(−3, −1)
y
2
4
-2
AVOID COMMON ERRORS
10. A(−3, −3), B(1, 3), C(3, −1); y-axis
P(−2, 3), Q(4, 3), R(−1, 0), S(-4, 1); x-axis
9.
-4 L′
E′
2
E x
0
D′2
-2 G -4
D'(1, −1), E'(2, 3), F'(−1, 4), G'(−3, −1)
4F
Given that △A'B'C' is the image of △ABC under a reflection, draw the line of reflection. 13.
B
4 C 2
y ℓ C′
-2
A
2
A′
4
14. midpoint of _ AA' is (-3, -1). _ midpoint of BB' is (-1, _1). midpoint x of CC' is (0, 2).
B′
A′
-4 A
ℓ
-4
2 -2
A A′
0
2
-2 -4
midpoint of _ AA' is (-3, 1). midpoint of _ BB' is (-1, 0)._ x C′ midpoint of CC' 4 is (1, -1).
Exercise
-2 -4
ℓ
midpoint of _ AA' is (-2, 0)_ . midpoint of BB' is (2, 2). B′ x
B
ℓ
C -4
-2
0
4
C
y
A
x
2
C′
2
4
-2 -4 A′
B
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C′
0
_ midpoint of AA' is (-3, -3). _ midpoint of BB' is (-1, -1). _ midpoint of CC' is (2, 2).
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Depth of Knowledge (D.O.K.)
© Houghton Mifflin Harcourt Publishing Company
B′
-2
B
y 4 C
y B′
2
-4
15.
4
COMMON CORE
Mathematical Practices
22
2 Skills/Concepts
MP.3 Logic
23
2 Skills/Concepts
MP.6 Precision
24
2 Skills/Concepts
MP.2 Reasoning
25
2 Skills/Concepts
MP.6 Precision
26
3 Strategic Thinking
MP.4 Modeling
27
2 Skills/Concepts
MP.3 Logic
28
3 Strategic Thinking
MP.3 Logic
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17. Jamar is playing a video game. The object of the game is to roll a marble into a target. In the figure, the shaded rectangular area represents the video screen and the striped rectangle is a barrier. Because of the barrier, it is not possible to roll the marble M directly into the target T. At what point should Jamar aim the marble so that it will bounce off a wall and roll into the target?
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Reflections, or flips, are one of the three rigid
y
motions that students will study. Reflections may be considered the most basic of the three because the other two can be expressed in terms of reflections. In particular, every translation is a composition of reflections across parallel lines and every rotation is a composition of reflections across intersecting lines.
T
B
C
4 2 A 0
x M 2
4
6 D
Reflect point M across the edge of the screen to locate M'. The coordinates of M' are _ (−1, 0). Locate point X where M'T intersects the edge of the screen. The coordinates of point X are (0, 2). Jamar should aim for the point (0, 2) along the edge of the screen. 18. A trail designer is planning two trails that connect campsites A and B to a point on the river, line ℓ. She wants the total length of the trails to be as short as possible. At what point should the trails meet the river?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (b)©Comstock Images/Jupiterimages/Getty Images; (t)©Pavelk/Shutterstock
A
4
y B
2 -4
-2
0
ℓ 2
x 4
Reflect point _ B across the river to locate B' . The coordinates of B' are (1, −1). Locate point X where AB' intersects the river. The coordinates of point X are (−1, 1). The trail designer will have the trails meet the river at (−1, 1). Algebra In the figure, point K is the image of point J under a reflection across line ℓ. Find each of the following. 19. JM
_ Since line ℓ bisects JK, JM = MK.
2x + 4 = 4x - 6; 5 = x
JM = 2x + 4 = 2(5) + 4 = 14
20. y
ℓ
K (3y - 30)°
4x - 6
M
2x + 4 Since line ℓ is J _ perpendicular to JK, each_ angle formed by the intersection of line ℓ and JK measures 90°.
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21. Make a Prediction Each y 4 time Jenny presses the tab key on her keyboard, the 2 software reflects the logo x she is designing across the -4 -2 0 2 4 x-axis. Jenny’s cat steps on the keyboard and presses -2 the tab key 25 times. In which quadrant does the logo end up? Explain. When the tab key is pressed twice, the logo is reflected into Quadrant III and then reflected back to its original position in Quadrant II. So after the tab key is pressed 24 times, the logo is in its original position. When the tab key is pressed for the 25th time, the logo is reflected across the x-axis into Quadrant III. 22. Multi-Step Write the equation of the line of reflection. A
4 2 B
y
ℓ A′
B′
-2
23. Communicate Mathematical Ideas The figure shows rectangle PQRS and its image after a reflection across the y-axis. A student said that PQRS could also be mapped to its image using the translation (x, y) → (x + 6, y). Do you agree? Explain why or why not.
x 2
P
4
Q 4
y
Q′
P′
2
C -4 C′
)
_ -1 + 3 4 + 2 midpoint of AA' = _, _ = (1, 3) 2 2 _ -4 + 0 -2 + -4 midpoint of CC' = _, _ = (-2, -3) 2 2
)
3 - (-3) 6 (−2, −3) and (1, 3); m = ______ = _ = 2 3 1 - (-2) y = 2x + b; 3 = 2(1) + b; 1 = b; y = 2x + 1.
S
R-2
R′
-4
No; the translation would move rectangle PQRS into the same position as rectangle P′Q′R′S′, but the corresponding vertices would not be in the same locations.
24. Which of the following transformations map △ABC to a triangle that intersects the x-axis? Select all that apply. A. (x, y) → (-x, y) B. (x, y) → (x, -y) C. (x, y) → (y, x)
S′
5
y
D. (x, y) → (-y, -x)
E. (x, y) → (x, y + 1)
A. A′(-1, -3), B′(-2, 1), C′(-4, -1); A′B′ intersects x-axis. B. A′(1, 3), B′(2, -1), C′(4, 1); A′B′ intersects the x-axis.
C. A′(-3, 1), B′(1, 2), C′(-1, 4); no side intersects the x-axis.
D. A′(3, -1), B′(-1, -2), C′(1, -4); no side intersects the x-axis.
B -5
x C
5
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jim O Donnell/Alamy
( (
x
A -5
E. A′(1, -2), B′(2, 2), C′(4, 0); A′B′ intersects the x-axis. Module 17
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JOURNAL
H.O.T. Focus on Higher Order Thinking
25. Explain _ _the Error _ △M′N′P ′ is the image of △MNP. Casey draws MM ′, NN′, and PP ′. Then she finds the midpoint of each segment and draws line ℓ through the midpoints. She claims that line ℓ is the line of reflection. Do you agree? Explain _ _ _ No; line ℓ is not perpendicular to MM′, NN′, and PP′ so it cannot be the line of reflection. There is a translation required in addition to a reflection to map △MNP to △M′N′P′.
Have students list some everyday examples of reflections they have seen (such as reflections in water or in mirrors and windows) and then describe how a reflection is like the original object and how it is different.
5
y
N -5
P
M
x ℓ
M′
5
P′ -5 N′
26. Draw Conclusions Plot the images of points D, E, F, and G after a reflection across the line y = 2. Then write an algebraic rule for the reflection.
The reflection maps points as follows: D(-3, 3) → D’(-3, 1), E(-1, 2) → E’(-1, 2), F(2, 0) → F’(2, 4), G(4, 4) → G’(4, 0). The x-coordinate is unchanged and the y-coordinate is subtracted from 4. The rule is (x, y) → (x, 4 - y).
5 D
-5
y F′
G
F
G′
E E′
D′
x 5
-5
© Houghton Mifflin Harcourt Publishing Company
27. Critique Reasoning Mayumi wants to draw the line of reflection for the reflection that maps △ABC to △A′B′C′. She claims that she just needs to draw the line through the points X and Y. Do you agree? Explain.
Yes; points X and Y are fixed under the reflection, so they must lie on the line of reflection. Since two points determine a line, the line _ of reflection is XY. 28. Justify Reasoning Point Q is the image of point P under a reflection across line ℓ. Point R lies on line ℓ. What type of triangle is △PQR? Justify your answer.
Isosceles triangle; _ since a reflection _is a rigid motion, it preserves distance. Since RQ is the image of RP, RQ = RP. Therefore, the triangle has two sides with the same length, so it is isosceles.
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X
B′
B Y
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Students may have difficulty understanding
In order to see the entire length of your body in a mirror, do you need a mirror that is as tall as you are? If not, what is the length of the shortest mirror you can use, and how should you position it on a wall?
the graph they are asked to draw in (a). Each point on the left side of the mirror should be the same distance from the y-axis as the corresponding point is from the x-axis. Most important in terms of understanding the reflection is that only the portion of the mirror between (0, 7.5) and (0, 3.5) is involved in producing the reflection.
a. Let the x-axis represent the floor and let the y-axis represent the wall on which the mirror hangs. Suppose the bottom of your feet are at F( 3, 0 ), your eyes are at E( 3, 7 ), and the top of your head is at H(3, 8). Plot these points and the points that represent their reflection images. (Hint: When you look in a mirror, your reflection appears to be as far behind the mirror as you are in front of it.) Draw the lines of sight from your eyes to the reflection of the top of your head and to the reflection of the bottom of your feet. Determine where these lines of sight intersect the mirror. b. Experiment by changing your distance from the mirror, the height of your eyes, and/or the height of the top of your head. Use your results to determine the length of the shortest mirror you can use and where it should be positioned on the wall so that you can see the entire length of your body in the mirror.
a.
H′ E′
8
y
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 The branch of physics called optics is the
H E
6
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Eric Camden/Houghton Mifflin Harcourt
4 2 F′ -4 -2
x
F 0
2
4
_ The lines of sight intersect the mirror _ at the midpoint of EH′, which is (0, 7.5), and at the midpoint of EF′, which is( 0, 3.5).
b. No matter what values you use for your distance from the mirror, the height of your eyes, and/or the height of the top of your head, the length of the shortest mirror that shows the entire length of your body is one-half your height. For example, in the figure from Part a, the viewer’s height is 8 units and the height of the shortest possible mirror is 7.5 - 3.5 = 4 units. The top of the mirror should be placed halfway between the top of your head and eye level. The bottom of the mirror should be placed halfway between eye level and the bottom of your feet.
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study of the properties and behavior of light. One of the fundamental principles of the field is that when a beam of light strikes a reflective surface, the angle of incidence (the angle of the incoming beam) is congruent to the angle of reflection.
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Students will need a protractor and a pocket mirror with a flat edge.
_ _ _ 1. Draw a line segment AB on a piece of paper and line segment CD meeting AB at _ D, forming ∠CDB. _ _ _ 2. Place the mirror along AB so you can see C'D, the reflection of CD. _ _ 3. Observing C'D and placing a straightedge under the mirror, draw DE, the _ extension of C'D. 4. Measure ∠CDB and ∠EDA. Suppose a light ray from C struck the mirror at D. Based on your results, at what angle do you think the ray would reflect off the mirror? an angle congruent to the angle at which the light ray struck the mirror
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Reflections 856
LESSON
17.3
Name
Rotations
Class
Date
17.3 Rotations Essential Question: How do you draw the image of a figure under a rotation?
Common Core Math Standards The student is expected to: COMMON CORE
Resource Locker
G-CO.A.4
Explore
Exploring Rotations
Develop definitions of rotations ... in terms of angles, circles, ... and line segments. Also G-CO.A.2, G-CO.A.5, G-CO.B.6
You can use geometry software or an online tool to explore rotations.
Mathematical Practices
A
Draw a triangle and label the vertices A, B, and C. Then draw a point P. Mark P as a center. This will allow you to rotate figures around point P.
B
Select △ABC and rotate it 90° around point P. Label the image of △ABC as △A′B ′C ′. Change the shape, size, or location of △ABC and notice how △A′B ′C ′ changes.
C
Draw ∠APA′, ∠BPB′, and ∠CPC ′. Measure these angles. What do you notice? Does this relationship remain true as you move point P? What happens if you change the size and shape of △ABC?
COMMON CORE
MP.5 Using Tools
Language Objective Students work in small groups or pairs to identify and label the transformation shown on a coordinate plane and if a rotation, identify the point of rotation.
ENGAGE Possible answer: To draw the image of a figure under a rotation of m° around point P, choose a vertex of the figure, for example, vertex A. Draw ¯ PA. Use a protractor to draw a ray that forms an angle of _ m° with PA. Use a ruler to mark point A' along the ray so that PA' = PA. Repeat the process with the other vertices of the figure. Connect the images of the vertices (A′, B′, etc.) to draw the image of the figure. If the figure is on a coordinate plane, use an algebraic rule to find the image of each vertex of the figure. Then connect the images of the vertices.
© Houghton Mifflin Harcourt Publishing Company
Essential Question: How do you draw the image of a figure under a rotation?
The measure of each angle is 90°; this remains true regardless of the location of point P or the size and shape of △ABC.
D
Measure the distance from A to P and the distance from A' to P. What do you notice? Does this relationship remain true as you move point P? What happens if you change the size and shape of △ABC?
AP = AP '; this remains true regardless of the location of point P or the size and shape
PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the motion of the minute hand of the clock with students. Then preview the Lesson Performance Task.
of △ABC. Module 17
be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction
Date Class Name
ions 17.3 Rotat
n? under a rotatio of a figure the image nts. Also do you draw and line segme ion: How circles, … of angles, … in terms of rotations p definitions G-CO.A.4 Develo .5, G-CO.B.6 G-CO.A G-CO.A.2, ations loring Rot ns. Exp rotatio e Explore tool to explor re or an online try softwa geome You can use
Quest Essential COMMON CORE
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y g Compan Publishin © Houghto
n Mifflin
Harcour t
? Does this you notice . What do size and these angles ns if you change the ′. Measure happe and ∠CPC P or point P? What ′, ∠BPB′, ion of point as you move Draw ∠APA the locat remain true dless of relationship true regar ? remains 90°; this shape of △ABC angle is ure of each The meas . ? Does of △ABC do you notice and shape A' to P. What change the size and the size from ce you the distan happens if to P and ce from A point P? What shape the distan you move size and Measure n true as P or the nship remai ion of point this relatio ? of the locat shape of △ABC regardless ins true Lesson 3 '; this rema AP = AP
.
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IN1_MNL
Watch for the hardcover student edition page numbers for this lesson.
d it 90° aroun and rotate as of △ABC Select △ABC the image or point P. Label e the shape, size, Chang △A′B ′C ′. notice how △ABC and location of changes. △A′B ′C ′
Module 17
Lesson 17.3
Resource Locker
s the vertice le and label a point P. Draw a triang C. Then draw will allow A, B, and P. a center. This Mark P as figures around point you to rotate
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Lesson 3
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gh "File info"
made throu
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Reflect
EXPLORE
1.
What can you conclude about the distance of a point and its image from the center of rotation? A point and its image are both the same distance from the center of rotation.
2.
What are the advantages of using geometry software or an online tool rather than tracing paper or a protractor and ruler to investigate rotations? Sample answer: Software or an online tool makes it easy to observe the effect of changing
Exploring Rotations INTEGRATE TECHNOLOGY
the shape or location of the preimage or changing the location of the center of rotation.
Explain 1
If time permits, students can use the software to experiment with different angles of rotation. In particular, ask students to investigate a 360° angle of rotation. Students should discover that the image of a figure after a 360° rotation coincides exactly with the preimage. Point out that this means a 360° rotation is equivalent to a 0° rotation. Students may also explore angles of rotation greater than 360°. In this case, students should conclude that an equivalent rotation can be found by subtracting 360°(or multiples of 360°) from the angle of rotation.
Rotating Figures Using a Ruler and Protractor
A rotation is a transformation around point P, the center of rotation, such that the following is true.
• Every point and its image are the same distance from P. • All angles with vertex P formed by a point and its image have the same measure. This angle measure is the angle of rotation.
In the figure, the center of rotation is point P and the angle of rotation is 110°. Example 1
A′
Draw the image of the triangle after the given rotation.
Counterclockwise rotation of 150° around point P
110° P
C
A
QUESTIONING STRATEGIES
B
A
In what direction does the software rotate figures? How could you use the software to produce a 90° clockwise rotation? Counterclockwise; enter 270° as the angle of rotation.
P © Houghton Mifflin Harcourt Publishing Company
_ _ Step 1 Draw PA. Then use a protractor to draw a ray that forms a 150° angle with PA.
C
A
B
P
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EXPLAIN 1 Rotating Figures Using a Ruler and Protractor INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Encourage students to use their knowledge of
Lesson 3
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M17L3 858
Learning Progressions In this lesson, students extend the informal concept of a rotation as a “turn” to a more precise definition. Rotations are one of the three rigid motions that students study in this module (translations and reflections are the other two). Rotations are somewhat more difficult to draw than the other rigid motions and predicting the effect of a rotation may be more difficult for students than predicting the effect of a reflection or a translation. Geometry software is a useful tool for investigating rotations. Students will need a solid understanding of transformations, including rotations, when they combine transformations to solve real-world problems.
4/19/14 10:49 AM
right angles to visualize rotations. Remind students that a 90° rotation is a quarter turn; a 45° rotation is half that. For example, suggest students visualize what is approximately a triangle after a rotation of 40° around P.
Rotations
858
Step 2 Use a ruler to mark point A′ along the ray so that PA′ = PA.
QUESTIONING STRATEGIES How can you use tracing paper to check your construction? Trace the figure; place a pencil’s point on P, and rotate the paper counterclockwise for the given angle of rotation. The traced version of the figure should lie on top of the rotated figure.
C
B
A
A′ P
Step 3 Repeat Steps 1 and 2 for points B and C to locate points B ′ and C ′. Connect points A′, B ′, and C ′ to draw △A′B ′C ′. C B′ C′
A
A′
B
P
B
Clockwise rotation of 75° around point Q _ _ Step 1 Draw QD. Use a protractor to draw a ray forming a clockwise 75° angle with QD. Step 2 Use a ruler to mark point D ′ along the ray so that QD ′ = QD. Step 3 Repeat Steps1 and 2 for points E and F to locate points E ′ and F ′. Connect points D ′, E ′, and F ′ to draw △D ′E ′F ′. F E
© Houghton Mifflin Harcourt Publishing Company
D
Q
E′
D′
F′
Reflect
3.
How could you use tracing paper to draw the image of △ABC in Part A? Put a piece of tracing paper on the page and trace △ABC and point P. Put the point of
a pencil on point P and use a protractor to rotate the tracing paper counterclockwise 150°. Trace over △ABC in the new location, pressing firmly to make an impression on the page below.
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Lesson 3
COLLABORATIVE LEARNING IN1_MNLESE389762_U7M17L3 859
Small Group Activity Have students work in small groups to write together a description of the similarities and differences they observe among the three transformations: translations, reflections, and rotations. Sample answer: All three transformations preserve the size and shape of the original figure. Each transformation uses a different geometric object (vector, line, or point) to perform the transformation. Translations always preserve the orientation of the original figure, while reflections and rotations may alter the orientation.
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Your Turn
EXPLAIN 2
Draw the image of the triangle after the given rotation. 4.
Counterclockwise rotation of 40° around point P
5.
Clockwise rotation of 125° around point Q
Drawing Rotations on a Coordinate Plane
T′
J′ L′
U′
K′
L
K
P
J
S
T
Explain 2
QUESTIONING STRATEGIES
S′
U
How can you predict the quadrant in which the image of the quadrilateral will lie? Every 90° of rotation moves the preimage around the origin by one quadrant, so a 270° rotation moves the preimage from Quadrant I to Quadrant IV.
Q
Drawing Rotations on a Coordinate Plane
You can rotate a figure by more than 180°. The diagram shows counterclockwise rotations of 120°, 240°, and 300°. Note that a rotation of 360° brings a figure back to its starting location.
A′
When no direction is specified, you can assume that a rotation is counterclockwise. Also, a counterclockwise rotation of x° is the same as a clockwise rotation of (360 - x)°.
240°
How can you use the rule for rotation to show that the origin is fixed under the rotation? The rule is (x, y) → (y, -x), so (0, 0) → (0, 0), which shows that the origin is fixed.
y
A x 120°
300°
The table summarizes rules for rotations on a coordinate plane.
A′′
A′′′
Rules for Rotations Around the Origin on a Coordinate Plane 90° rotation counterclockwise 180° rotation 270° rotation counterclockwise
Example 2
(x, y) → (-x, -y)
Some students may confuse the direction of a rotation (clockwise or counterclockwise). Remind students that the direction is assumed to be counterclockwise unless otherwise stated. Associate this default direction with the way the quadrants are numbered.
(x, y) → (y, -x) (x, y) → (y, x)
© Houghton Mifflin Harcourt Publishing Company
360° rotation
AVOID COMMON ERRORS
(x, y) → (-y, x)
Draw the image of the figure under the given rotation.
Quadrilateral ABCD; 270°
4
The rotation image of (x, y) is (y, −x).
A
Find the coordinates of the vertices of the image. A(0, 2) → A′(2, 0)
-4
B(1, 4) → B'(4, -1)
yB
-2
C x
D 0
2
4
-2
COMMUNICATING MATH
-4
C(4, 2) → C '(2, -4)
D(3, 1) → D'(1, -3)
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Students analyze pictures of preimages and images, and discuss what kind of transformation is shown. The group must agree before labeling each picture. If a transformation is identified as a rotation, the group must determine the point of rotation. Each picture should be labeled, and this sentence completed: “This shows a (translation/reflection/ ”. Provide key terms to rotation) because help students complete the statement.
Rotations 860
Predict the quadrant in which the image will lie. Since quadrilateral ABCD lies in Quadrant I and the quadrilateral is rotated counterclockwise by 270°, the image will lie in Quadrant IV. Plot A′, B′, C ′, and D′ to graph the image.
-4
-2
A
C
0
A′ D x 2 4B′
-2 -4
△KLM; 180°
B
(
)
Find the coordinates of the vertices of the image.
(
(
L(4, -1) → L' -4 , 1
(
M(1, -4) → M' -1 , 4
) )
D′ C′
M′ y 4
The rotation image of (x, y) is -x , -y .
K(2, -1) → K' -2 , 1
yB
4
L′ -4
K′ -2
2 x 0
-4
)
K2
4L
M
Predict the quadrant in which the image will lie. Since △KLM lies in Quadrant
IV
and
the triangle is rotated by 180°, the image will lie in Quadrant II . Plot K', L', and M' to graph the image.
© Houghton Mifflin Harcourt Publishing Company
Reflect
6.
Discussion Suppose you rotate quadrilateral ABCD by 810°. In which quadrant will the image lie? Explain. Quadrant II; the quadrilateral ABCD is in Quadrant 1. Every rotation of 360° brings the quadrilateral back to Quadrant I, and since 810° = 360° + 360° + 90°, the 810° rotation is equivalent to a 90° rotation. This maps the quadrilateral to Quadrant II.
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Lesson 3
LANGUAGE SUPPORT IN1_MNLESE389762_U7M17L3 861
The words rotation and transformation (as well as function and notation) are all cognates with Spanish. They contain the same Latin root and have similar spellings and identical meanings. Point out that all these words in English end with –tion, and in Spanish they all end with –ción. This is a word pattern that may be useful to students who speak English and Spanish.
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Your Turn
EXPLAIN 3
Draw the image of the figure under the given rotation. 7.
△PQR; 90°
8. Q′ 4 P′ 2
-4
-2
y
R′ P
0
Example 3
Q x
F′
D′
2 E′
D
-4
Specifying Rotation Angles
y
-2
QUESTIONING STRATEGIES
x 2
E
When a drawing of a rotation shows two figures, how can you tell which is the preimage and which is the image? The vertices of the image will have prime marks, for example, A’.
G
R
F
Specifying Rotation Angles
AVOID COMMON ERRORS
Find the angle of rotation and direction of rotation in the given figure. Point P is the center of rotation.
C′
B′
4
G′
2
-4
Explain 3
Quadrilateral DEFG; 270°
Some students may have difficulty identifying the direction of the rotation. Suggest that they visualize P as the center of a clock, with the minute hand pointing to a vertex on the preimage. As the minute hand moves to point at the corresponding vertex, which way is it moving (going the shortest way)?
D′
A′
B
C
A
D
C′
Draw segments from the center of rotation to a vertex and to the image of the vertex.
B′
A′
Measure the angle formed by the segments. The angle measure is 80°. Compare the locations of the preimage and image to find the direction of the rotation. P
The rotation is 80° counterclockwise.
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862
D′
B
C
A
D
© Houghton Mifflin Harcourt Publishing Company
P
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Rotations 862
R
S
B
Q
S′ Q′
P
R′
Draw segments from the center of rotation to a vertex and to the image of the vertex. Measure the angle formed by the segments. The angle measure is 135 °. The rotation is 135 ° (clockwise/counterclockwise). Reflect
9.
Discussion Does it matter which points you choose when you draw segments from the center of rotation to points of the preimage and image? Explain. No, as long as the points are corresponding points (i.e., a point and its image), the angle of rotation will be the same.
10. In Part A, is a different angle of rotation and direction possible? Explain. Yes; a rotation of 80° counterclockwise is equivalent to a rotation of 280° clockwise.
Your Turn
© Houghton Mifflin Harcourt Publishing Company
Find the angle of rotation and direction of rotation in the given figure. Point P is the center of rotation. P
11.
K L N N′ M
K′
M′ L′
The transformation is a 70° clockwise rotation.
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Elaborate
ELABORATE
12. If you are given a figure, a center of rotation, and an angle of rotation, what steps can you use to draw the image of the figure under the rotation? Sample answer: Draw a segment from the center of rotation, P, to one vertex of the _ figure, A. Use a protractor to draw a ray that forms an angle with PA that is equal to the
QUESTIONING STRATEGIES
_ Given a line segment WP, describe how you _ would draw WP under a rotation of 120° around P. Draw a ray from P that forms a 120° angle with the segment. Mark point W′ along this ray such that WP = W′P.
angle of rotation. Use a ruler to mark a point along the ray so that PA' = PA. Repeat the process with the other vertices of the figure. Connect the images of the vertices to draw the image of the figure. 13. Suppose you are given △DEF, △D'E 'F ', and point P. What are two different ways to prove that a rotation around point P cannot be used to map △DEF to △D'E 'F '? (1) Show that PD ≠ PD', PE ≠ PE ', or PF ≠ PF '. (2) Show that m∠DPD' ≠ m∠EPE ',
m∠EPE ≠ m∠FPF ', or m∠DPD' ≠ m∠FPF '.
SUMMARIZE THE LESSON
14. Essential Question Check-In How do you draw the image of a figure under a counterclockwise rotation of 90° around the origin? For each vertex of the figure, use the rule (x, y) → (-y, x) to find the coordinates of the
How do you draw the image of a figure under a clockwise rotation of 90° around the origin? For each vertex of the figure, use the rule (x, y) → (y,-x) to find the coordinates of the image of the vertex. Plot the images of the vertices; then connect these points to draw the image of the figure.
image of the vertex. Plot the images of the vertices, then connect these points to draw the image of the figure.
Evaluate: Homework and Practice 1.
• Online Homework • Hints and Help • Extra Practice
Which three angles must have the same measure? What is the measure of these angles?
∠SPS ', ∠TST ', and ∠USU '; all three angles measure 115 ° since this is the amount by which the triangle was rotated around the point.
Module 17
Exercise
IN1_MNLESE389762_U7M17L3 864
ASSIGNMENT GUIDE
Lesson 3
864
Depth of Knowledge (D.O.K.)
EVALUATE © Houghton Mifflin Harcourt Publishing Company
Alberto uses geometry software to draw △STU and point P, as shown. He marks P as a center and uses the software to rotate △STU 115° around point P. He labels the image of △STU as △S'T 'U '.
COMMON CORE
Mathematical Practices
1–4
1 Recall of Information
MP.5 Using Tools
5–8
1 Recall of Information
MP.4 Modeling
9–10
1 Recall of Information
MP.5 Using Tools
11–13
2 Skills/Concepts
MP.2 Reasoning
14
2 Skills/Concepts
MP.3 Logic
15
2 Skills/Concepts
MP.2 Reasoning
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Concepts and Skills
Practice
Explore Exploring Rotations
Exercise 1
Example 1 Rotations Figures Using a Ruler and Protractor
Exercises 2–4
Example 2 Drawing Rotations on a Coordinate Plane
Exercises 5–8
Example 3 Specifying Rotation Angles
Exercises 9–10
Rotations 864
Draw the image of the triangle after the given rotation.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Remind students that in mapping a figure
2.
Counterclockwise rotation of 30 ° around point P
3.
Clockwise rotation of 55 ° around point J M′
B′
L′
onto itself, the center of rotation is inside the figure. If students are confused, show a square rotating around a point at its center and a square rotating around a point outside the square. Help students verbalize the difference.
C′ A′ B A
P
K′ C
J
L
K
M
CONNECT VOCABULARY Have students complete a chart like the following vocabulary chart, filling in the blank areas with pictures and words to describe the transformation.
4.
Counterclockwise rotation of 90° around point P P F′
Transformations
D
Reflection
Rotation
Define or describe:
Define or describe:
Define or describe:
Draw an example:
Draw an example:
Draw an example:
F
E
D′
© Houghton Mifflin Harcourt Publishing Company
Translation
Draw the image of the figure under the given rotation. 5.
△ABC; 270° A C -4
6. 4 2
B -2 0
y
Exercise
IN1_MNLESE389762_U7M17L3 865
Lesson 17.3
△RST; 90°
C′
4 R′ A′
B′
x 2
-4
4
-2
y
T′
2 0
-2
-2
-4
-4
Module 17
865
E′
R 2
S
x 4
T
Lesson 3
865
Depth of Knowledge (D.O.K.)
S′
COMMON CORE
Mathematical Practices
16
2 Skills/Concepts
MP.2 Reasoning
17–20
2 Skills/Concepts
MP.4 Modeling
21
2 Skills/Concepts
MP.2 Reasoning
22
3 Strategic Thinking
MP.5 Using Tools
23–24
3 Strategic Thinking
MP.3 Logic
25
3 Strategic Thinking
MP.1 Problem Solving
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Quadrilateral EFGH; 180°
7.
y
4
G′
P′
E′
0
2
-4
x
-2 0 Q R′
P
G
-4
Some students may rotate a figure around its center or around one of its vertices, not around point P. Reread the instructions together. Ask students to explain the differences between rotating a figure around an exterior point, around a point on the figure, and around an interior point.
2 Q′
x 4H
-2
AVOID COMMON ERRORS
y
4
S′
2
F
Quadrilateral PQRS; 270°
F′
H′ -4 E -2
8.
2 R
-4
S
4
Find the angle of rotation and direction of rotation in the given figure. Point P is the center of rotation. Z′
9.
X′
V
10.
Y′
U W
Z
P X
P
Y
W′ U′ V′
The rotation is 50° counterclockwise.
The rotation is 180°.
Write an algebraic rule for the rotation shown. Then describe the transformation in words. 4 E D
2
y
12.
G′
4 S
R
F′
y
2 x
-4
-2
G
0
2
F E′ -4
x -4 S′
4 D′
0
T R′
-4
2
4 T′
D(-4, 2) → D'(4, -2), E(-1, 3) → E '(1, -3),
R(-4, 3) → R'(-3, -4), S(-1, 3) →
so the image of (x, y) is (-x, -y). The rule
the image of (x, y) is (-y, x). The rule
F(-1, -3) → F '(1, 3), G(-3, -4) → G '(3, 4)
is (x, y) → (-x, -y) and the transformation
IN1_MNLESE389762_U7M17L3 866
S '(-3, -1), T(-3, -3) → T '(3, -3), so is (x, y) → (-y, x) and the transformation
is a counterclockwise rotation of 90°.
is a rotation of 180°. Module 17
© Houghton Mifflin Harcourt Publishing Company
11.
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Rotations 866
13. Vanessa used geometry software to apply a transformation to △ABC, as shown. According to the software, m∠APA' = m∠BPB' = m∠CPC '. Vanessa said this means the transformation must be a rotation. Do you agree? Explain. No; according to the definition of a rotation, B′ every point and its image must be the same C′
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Discuss the fact that every rotation can be
distance from P, and that is not the case in the
expressed as a composition of reflections across intersecting lines.
A′
B
P
given figure.
A C
14. Make a Prediction In which quadrant will the image of △FGH lie after a counterclockwise rotation of 1980°? Explain how you made your prediction. Quadrant I; a rotation of 360° brings the figure back to y 4 its original location, so you can subtract multiples of 2 x -4
F
0
2
4
1980° - 1800° = 180°, so the rotation is equivalent to a rotation of 180°, which maps △FGH to Quadrant I.
-2
G
360° from the angle of rotation. 360° × 5 = 1800° and
H
15. Critical Thinking The figure shows the image of △MNP after a counterclockwise rotation of 270°. Draw and label △MNP. The rule for the rotation is (x, y) → (y, -x). P y M′ 4 M' has coordinates (2, 4), so the coordinates of M are (-4, 2). © Houghton Mifflin Harcourt Publishing Company
M
-4
2 N
-2
0
N′
P′x
2
4
N ' has coordinates (2, 1), so the coordinates of N are (−1, 2).
P ' has coordinates (4, 1), so the coordinates of P are (−1, 4).
-2 -4
16. Multi-Step Write the equation of the image of line ℓ after a clockwise rotation of 90°. (Hint: To find the image of line ℓ, choose two or more points on the line and find the images of the points.) Line ℓ passes through A(2, 0) and B(0, −1). The rule for a clockwise y 4 rotation of 90° is (x, y) → (-y, x), so A(2, 0) → A'(0, 2) and. 2
-4
-2
0
ℓ 2
x 4
0-2 The line through A' and B ' has slope ____ = -2 and y-intercept 2, 1-0
so the equation of the line is y = −2x + 2.
-2 -4
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DIFFERENTIATE INSTRUCTION IN1_MNLESE389762_U7M17L3 867
Modeling To reinforce the meaning of rotation, show some examples of rotations that might be familiar to students—for example, the turn of a steering wheel or Earth’s orbit around the sun. Ask students to give other examples of turns or rotations in the real world, such as the motion of a DVD in a player, or of a doorknob. Discuss what all of the motions have in common. Help students see that all involve moving points around a fixed point.
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17. A Ferris wheel has 20 cars that are equally spaced around the circumference of the wheel. The wheel rotates so that the car at the bottom of the ride is replaced by the next car. By how many degrees does the wheel rotate? 18°; there are 360° in a complete rotation
and there are 20 equally-spaced cars, so the amount of rotation is 360° ÷ 20 = 18°. 18. The Skylon Tower, in Niagara Falls, Canada, has a revolving restaurant 775 feet above the falls. The restaurant makes a complete revolution once every hour. While a visitor was at the tower, the restaurant rotated through 135°. How long was the visitor at the tower? Set up a proportion. 135 x = 360 60 x = 22.5
_ _
The visitor was at the tower for 22.5 minutes. 19. Amani plans to use drawing software to make the design shown here. She starts by drawing Triangle 1. Explain how she can finish the design using rotations. Possible answer: Starting with triangle 1, rotate clockwise 60º around
2
1
the vertex at the center of the hexagon. Repeat the process using each
5
3 4
successive image as a pre-image. 20. An animator is drawing a scene in which a ladybug moves around three mushrooms. The figure shows the starting position of the ladybug. The animator rotates the ladybug 180° around mushroom A, then 180° around mushroom B, and finally 180° around mushroom C. What are the final coordinates of the ladybug? (2, -4); the 180° rotation around A moves the ladybug from
4
y
A B -4
(-4, 2) to (0, 2); the 180° rotation around B moves the ladybug
from (0, 2) to (4, 0); the 180° rotation around C moves the
x
0
4 C
-4
ladybug from (4, 0) to (2, -4).
21. Determine whether each statement about the rotation (x, y) → (y, -x) is true or false. Select the correct answer for each lettered part. a. Every point in Quadrant I is mapped to a point in Quadrant II. True b. Points on the x-axis are mapped to points on the y-axis.
False
True
False
c. The origin is a fixed point True under the rotation.
False
Module 17
IN1_MNLESE389762_U7M17L3 868
868
d. The rotation has the same effect as a 90° clockwise rotation.
True
False
e. The angle of rotation is 180°.
True
False
f. A point on the line y = x is mapped to another point on the line y = x.
True
False
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t)©Yuttasak Jannarong/Shutterstock; (b)©Joe Sohm/Visions of America/Photodisc/Getty Images
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Rotations 868
spaced around the tates so that the car e next car. By how
JOURNAL
H.O.T. Focus on Higher Order Thinking
22. Communicate Mathematical Ideas Suppose you are given a figure and a center of rotation P. Describe two different ways you can use a ruler and protractor to draw the image of the figure after a 210° counterclockwise rotation around P. Sample answer: Make a table that shows the quadrant the triangle will lie in for various
Have students list some everyday examples of rotations and how they are used, and then describe how a rotation is like the original object and how it is different.
values of n. 23. Explain the Error Kevin drew the image of △ABC after a rotation of 85° around point P. Explain how you can tell from the figure that he made an error. Describe the error. Possible answer: △A'B'C ' should be rotated so that B' is at
da, has a revolving taurant makes a ile a visitor was at 135°. How long
C′ B′
A′ C
the top of the figure. After correctly locating the image of 85°
point A, Kevin translated the figure rather than rotating it.
B
A
24. Critique Reasoning Isabella said that all points turn around the center of rotation by the same angle, so all points move the same distance under a rotation. Do you agree with Isabella’s statement? Explain. No; although all points rotate through the same angle, points closer to the center of
make the design shown here. She starts by inish the design using rotations.
rotation move a shorter distance than points father from the center of rotation. 25. Look for a Pattern Isaiah uses software to draw △DEF as shown. Each time he presses the left arrow key, the software rotates the figure on the screen 90° counterclockwise. Explain how Isaiah can determine which quadrant the triangle will lie in if he presses the left arrow key n times. Possible answer: (1) Use the ruler and protractor to draw a
a ladybug moves around three g position of the ladybug. The mushroom A, then 180° around shroom C. What are the final
4
y
F
E
2 -4
150° clockwise rotation of the figure. (2) First draw a 180°
-2
x
D 0
2
4
-2
rotation of the figure. Then draw a 30° counterclockwise
the rotation (x, y) → (y, -x) is true or ttered part.
False
False False
d. The rotation has the same effect as a 90° clockwise rotation.
True
False
e. The angle of rotation is 180°.
True
False
f. A point on the line y = x is mapped to another point on the line y = x.
True
False
© Houghton Mifflin Harcourt Publishing Company
rotation of the image. n Quadrant
3 IV
4 I
5 II
6 III
A remainder of 0 → QI
A remainder of 2 → QIII
A remainder of 1 → QII
A remainder of 3 → QIV
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2 III
7 IV
8 I
The remainder after dividing n by 4 defines a pattern for the table.
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Ask students to look at the table they made
A tourist in London looks up at the clock in Big Ben tower and finds that it is exactly 8:00. When she looks up at the clock later, it is exactly 8:10.
for the angles of rotation of the hour hand and minute hand of Big Ben. Ask: “How, if at all, would the table change if you made a similar table for a small pocket watch?” Explain your reasoning. The table would not change. A complete rotation around the face of any clock or any circle, no matter how big or how small, equals 360°. The values in the table are based on that fact, not on the size of the circle.
a. Through what angle of rotation did the minute hand turn? Through what angle of rotation did the hour hand turn? b. Make a table that shows different amounts of time, from 5 minutes to 60 minutes, in 5-minute increments. For each number of minutes, provide the angle of rotation for the minute hand of a clock and the angle of rotation for the hour hand of a clock.
a. A complete rotation around the face of the clock is 360°. The face of the clock is divided into 12 equal parts, each representing 5-minute intervals. So the angle of rotation of the minute hand is 360° ÷ 12 = 30° for every 5 minutes. During an interval of 10 minutes, the angle of rotation of the minute hand is 60°. In one hour, the hour hand moves from one number on the face of the clock to the next, which is an angle of rotation of 30°. Since an hour is 60 minutes, a 1 of this angle of rotation, or 5°. 10-minute interval represents _ 6
QUESTIONING STRATEGIES Have students refer to their angles of rotation tables.
b. As above, the angle of rotation of the minute hand is 30° for every 5 minutes. The angle of rotation of the hour hand is 5° for every 10 minutes, or 2.5° for every 5 minutes. Use these values to complete the table below. Amount of Time (min)
Angle of Rotation, Hour Hand
5
30°
2.5°
10
60°
5.0°
15
90°
7.5°
20
120°
10.0°
25
150°
12.5°
30
180°
15.0°
35
210°
17.5°
40
240°
20.0°
45
270°
22.5°
50
300°
25.0°
55
330°
27.5°
60
360°
30.0°
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© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Andrew Ward/Life File/Photodisc/Getty Images
Angle of Rotation, Minute Hand
When the amount of time doubles, how are the angles of rotation affected? They double. When the amount of time increases by 15 minutes, how are the angles of rotation affected? The angle of rotation of the minute hand increases by 90°. The angle of rotation of the hour hand increases by 7.5°. If you know the angle of rotation of the minute hand, how can you find the angle of rotation of the hour hand? Divide the angle of rotation of the minute hand by 12.
Lesson 3
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Big Ben isn’t the biggest clock in the world. Have students conduct research to find five clocks bigger than Big Ben. Ask them to present their findings as five congruent circles drawn to scale, with labels detailing the sizes of the clocks, and giving interesting facts about each. Students should include information about the angles formed on each clock as the time changes. Have students consider how often during a 12-hour period the hour and minute hands of each clock are at a 180° degree angle to each other. They can then calculate or estimate one or more times that this would occur, other than at 6 o’clock.
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
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LESSON
17.4
Name
Investigating Symmetry
Essential Question: How do you determine whether a figure has line symmetry or rotational symmetry?
1
The student is expected to:
Explore 1
Resource Locker
Identifying Line Symmetry
A figure has symmetry if a rigid motion exists that maps the figure onto itself. A figure has line symmetry (or reflectional symmetry) if a reflection maps the figure onto itself. Each of these lines of reflection is called a line of symmetry.
G-CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Line of symmetry
You can use paper folding to determine whether a figure has line symmetry.
Mathematical Practices COMMON CORE
Date
17.4 Investigating Symmetry
Common Core Math Standards COMMON CORE
Class
Trace the figure on a piece of tracing paper.
If the figure can be folded along a straight line so that one half of the figure exactly matches the other half, the figure has line symmetry. The crease is the line of symmetry. Place your shape against the original figure to check that each crease is a line of symmetry.
Sketch any lines of symmetry on the figure.
MP.7 Using Structure
Language Objective Have students work with a partner to give clues about a figure, and identify whether figures have line symmetry, rotational symmetry, or both and draw the line(s) of symmetry.
Essential Question: How do you determine whether a figure has line symmetry or rotational symmetry? Possible answer: To identify line symmetry, look for a line of reflection, which is a line that divides the figure into mirror-image halves. To identify rotational symmetry, think of the figure rotating around its center. The figure has rotational symmetry if a rotation of at most 180° produces the original figure.
© Houghton Mifflin Harcourt Publishing Company
ENGAGE
The figure has
one
line of symmetry.
PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo of the flower with students. Consider whether you could turn the flower and have it still appear the same. Then preview the Lesson Performance Task.
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Draw the lines of symmetry, if any, on each figure and tell the total number of lines of symmetry each figure has.
EXPLORE 1 Identifying Line Symmetry
Figure How many lines of symmetry?
two
zero
one
INTEGRATE TECHNOLOGY
Reflect
1.
Have students use geometry software or cut out figures to examine the symmetry of regular polygons. Then have them use inductive reasoning to make conjectures about the number of lines of symmetry a regular n-gon has. n lines of symmetry
What do you have to know about any segments and angles in a figure to decide whether the figure has line symmetry? Pairs of segments in the figure must have the same length and pairs of angles must have
the same measure, so that one half of the figure will coincide with the other half when the figure is folded across a line of symmetry. 2. 3.
What figure has an infinite number of lines of symmetry? A circle
QUESTIONING STRATEGIES
Discussion A figure undergoes a rigid motion, such as a rotation. If the figure has line symmetry, does the image of the figure have line symmetry as well? Give an example. Yes. The line of symmetry also undergoes the rigid motion. For example, if the L-shape in
What are the three rigid motions explained in this module? What does a rigid motion transformation preserve? translation, reflection, rotation; shape and size
Step D is rotated into a V-shape, the line of symmetry is rotated the same way.
1
Explore 2
Identifying Rotational Symmetry
A figure has rotational symmetry if a rotation maps the figure onto itself. The angle of rotational symmetry, which is greater than 0° but less than or equal to 180°, is the smallest angle of rotation that maps a figure onto itself.
Angle of rotational symmetry: 72°
Trace the figure onto tracing paper. Hold the center of the traced figure against the original figure with your pencil. Rotate the traced figure counterclockwise until it coincides again with the original figure beneath.
By how many degrees did you rotate the figure?
EXPLORE 2 Identifying Rotational Symmetry
© Houghton Mifflin Harcourt Publishing Company
An angle of rotational symmetry is a fractional part of 360°. Notice that 360° every time the 5-pointed star rotates ____ = 72°, the star coincides with 5 itself. The angles of rotation for the star are 72°, 144°, 216°, and 288°. If a copy of the figure rotates to exactly match the original, the figure has rotational symmetry.
INTEGRATE TECHNOLOGY Ask students to discuss the pros and cons of using geometry software to investigate properties of rotations and symmetry. Be sure students recognize that such software has the advantage of making it easy to change parameters (such as the angle of rotation) so that they can observe the effects of the changes.
120°
What are all the angles of rotation? 120°, 240° Module 17
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QUESTIONING STRATEGIES
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Math Background
A wallpaper pattern is a planar repeating pattern. Mathematicians classify wallpaper patterns based on the symmetries they exhibit, for example, only translation symmetry; translation and reflection symmetry; or all these plus rotational symmetry. Every wallpaper pattern can be classified by identifying its symmetries. Surprisingly, there are precisely 17 different classifications. That is, any repeating pattern that covers a plane can be reduced to one of 17 basic types. This unusual mathematical fact has had far-reaching applications in a number of fields, including chemistry and crystallography.
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When you are testing a figure to see if it has rotational symmetry, where is P, the center of rotation? at the center of the figure
Investigating Symmetry 872
EXPLAIN 1
Determine whether each figure has rotational symmetry. If so, identify all the angles of rotation less than 360°.
Describing Symmetries
Figure
INTEGRATE TECHNOLOGY
Angles of rotation less than 360°
Human faces appear to have symmetry, but most people’s faces aren’t perfectly symmetric. Photocopy a picture of a face onto two transparencies and cut each one down the center of the face. Flip the pieces of one transparency over and put the two left sides together and the two right sides together to create two different faces with perfect symmetry. Discuss with students how to tell if a figure has symmetry.
90°, 180°, 270°
none
180°
Reflect
4. 5.
What figure is mapped onto itself by a rotation of any angle? A circle
Discussion A figure is formed by line l and line m, which intersect at an angle of 60°. Does the figure have an angle of rotational symmetry of 60°? If not, what is the angle of rotational symmetry? No, the angle of rotational symmetry for the figure is 180°. A rotation of 60° about the intersection will only map one of the lines onto the other line.
Explain 1
Describing Symmetries
A figure may have line symmetry, rotational symmetry, both types of symmetry, or no symmetry. Example 1
QUESTIONING STRATEGIES How can you find the center point of a regular polygon? The center is the point that is equidistant from each vertex or corner.
Some students may think that any diagonal of a figure is a line of symmetry. Have them draw a rectangle that is not a square and one of the diagonals. Folding along this diagonal demonstrates that it is not a line of symmetry.
© Houghton Mifflin Harcourt Publishing Company
AVOID COMMON ERRORS
Describe the symmetry of each figure. Draw the lines of symmetry, name the angles of rotation, or both if the figure has both.
Step 1 Begin by finding the line symmetry of the figure. Look for matching halves of the figure. For example, you could fold the left half over the right half, and fold the top half over the bottom half. Draw one line of symmetry for each fold. Notice that the lines intersect at the center of the figure.
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Small Group Activity Have students identify common shapes that do not have line symmetry, for example, the capital letters F, G, and J. Have students name a letter of the alphabet with each type of symmetry: • a horizontal line of symmetry B C D E • a vertical line of symmetry A I M T U V W Y • two lines of symmetry H O X • rotational symmetry but not line symmetry N S Z • no symmetry F G J K L P Q R
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Step 2 Now look for other lines of symmetry. The two diagonals also describe matching halves. The figure has a total of 4 lines of symmetry.
CONNECT VOCABULARY Connect the idea of a reflection to a figure with line symmetry. If you identify the line of symmetry on the figure, and superimpose that line on the x- or y-axis on a coordinate plane, then the line of symmetry becomes the line of reflection, and you can see the image and preimage on either side.
Step 3 Next, look for rotational symmetry. Think of the figure rotated about its center until it matches its original position. The angle of rotational symmetry of this figure is __14 of 360°, or 90°. The other angles of rotation for the figure are the multiples of 90° that are less than 360°. So the angles of rotation are 90°, 180°, and 270°. Angle of rotational symmetry: 90°
Number of lines of symmetry: 4
Angles of rotation: 90°, 180°, 270°
B
© Houghton Mifflin Harcourt Publishing Company
Step 1 Look for lines of symmetry. One line divides the figure into left and right halves. Draw this line on the figure. Then draw similar lines that begin at the other vertices of the figure. Step 2 Now look for rotational symmetry. Think of the figure rotating about its center until it matches the original figure. It rotates around the circle by a fraction of which is
__1 3
120°
. Multiply by 360° to find the angle of rotation, . Find multiples of this angle to find other angles
of rotation. Number of lines of symmetry:
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Angles of rotation: 120°, 240°
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LANGUAGE SUPPORT IN1_MNLESE389762_U7M17L4 874
Connect Vocabulary
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In English and in Spanish, we usually add –s or –es to the end of a noun to form the plural, for example, triangles, points, figures. In English, some nouns are irregular and don’t follow that convention. The plural form of the noun vertex is vertices. Notice that the x becomes a c and then –es is added to form this plural. In Spanish, the same thing happens with words that end in z. The z becomes a c and then –es is added to form the plural.
Investigating Symmetry 874
Your Turn
ELABORATE
Describe the type of symmetry for each figure. Draw the lines of symmetry, name the angles of rotation, or both if the figure has both. 6.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 The number of angles of rotation less than
Figure ABCD
7. A
D
H
l
D
8.
1 Number of lines of symmetry: 90°, 180°, 270° Angles of rotation:
line, rotational
Figure KLNPR
9.
T L
R
Q
W
M P
Types of symmetry:
© Houghton Mifflin Harcourt Publishing Company
Figure TUVW
J K
S
line
Types of symmetry:
4 Number of lines of symmetry: 90°, 180°, 270° Angles of rotation:
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Give students pictures of figures or objects
How do you determine whether a figure has line symmetry or rotational symmetry? A figure has line symmetry if the figure can be reflected across a line so that the image coincides with the preimage. A figure has rotational symmetry if the figure can be rotated about a point by an angle greater than 0° and less than or equal to 180° so that the image coincides with the preimage.
G
B
Types of symmetry:
SUMMARIZE THE LESSON
F
E
360° is called the order of the rotational symmetry, so a square is of order 3 and an equilateral triangle is of order 2. A five-pointed star is of order 4.
that have line symmetry, rotational symmetry, both, or neither. Students get two sets of each figure. One set of pictures is placed face up between the pair of students and one set is face down. The first student draws a card, such as the picture of a square, and gives oral clues such as, “This figure has both rotational symmetry and line symmetry. Its angle of rotation is 90 degrees; it has 4 sides and 4 angles; and has 4 lines of symmetry.” The second student picks the picture that matches the clues. They switch roles and repeat the process.
Figure EFGHI
U V
Types of symmetry:
N
none 0
Number of lines of symmetry: Angles of rotation:
rotational
none
0 Number of lines of symmetry: Angles of rotation: 72°, 144°, 216° and 288°
Elaborate 10. How are the two types of symmetry alike? How are they different? Both types of symmetry show how a figure can be mapped onto itself by a rigid
motion. In line symmetry, the figure is mapped onto itself by reflection, and in rotational symmetry, the mapping is by rotation. 11. Essential Question Check-In How do you determine whether a figure has line symmetry or rotational symmetry? Possible answer: To identify line symmetry, look for a line of reflection that divides the
figure into mirror-image halves. To identify rotational symmetry, think of the figure rotating around its center. The figure has rotational symmetry if a rotation of at most 180° produces the original figure. Module 17
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Modeling Bring in books or suggest websites that show examples of mandalas. Have students find and describe examples of rotational symmetry in each. Have them create an original mandala.
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Evaluate: Homework and Practice
EVALUATE • Online Homework • Hints and Help • Extra Practice
Draw all the lines of symmetry for the figure, and give the number of lines of symmetry. If the figure has no line symmetry, write zero. 1.
2.
Lines of symmetry:
1
3.
Lines of symmetry:
8
Lines of symmetry:
ASSIGNMENT GUIDE
4
For the figures that have rotational symmetry, list the angles of rotation less than 360°. For figures without rotational symmetry, write “no rotational symmetry.” 4.
5.
6.
45°, 90°, 135°, 180°, Angles of Angles of no rotational symmetry rotation: 225°, 270°, 315° rotation:
Angles of no rotational symmetry rotation:
In the tile or basket shown, identify whether the pattern has line symmetry, rotational symmetry, both line and rotational symmetry, or no symmetry. 7.
For figure ABCDEF shown here, identify the image after each transformation _ described. For example, a reflection across AD has an image of figure AFEDCB. In the figure, all the sides are the same length and all the angles are the same measure. 9.
_ Reflection across CF
10. rotation of 240° clockwise, or 120° counterclockwise
Figure EDCBAF
Exercise
COMMON CORE
Mathematical Practices
1 Recall of Information
MP.4 Modeling
2
2 Skills/Concepts
MP.4 Modeling
1 Recall of Information
MP.4 Modeling
5
2 Skills/Concepts
MP.4 Modeling
6
1 Recall of Information
MP.4 Modeling
2 Skills/Concepts
MP.4 Modeling
7–8
C D
Explore 2 Identifying Rotational Symmetry
Exercises 4–6
Example 1 Describing Symmetries
Exercises 7–16
on translations. A pattern has translation symmetry if it can be translated along a vector so that the image coincides with the pre-image. Tiled floors may be examples of this.
Lesson 4
1
3–4
B
E
Exercises 1–3
11. reflection across the line that _ connects the midpoint _of BC and the midpoint of EF
876
Depth of Knowledge (D.O.K.)
A F
Explore 1 Identifying Line Symmetry
Figure DCBAFE
Figure CDEFAB
Module 17
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© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t)©costall/Shutterstock ; (b)©Cbenjasuwan/Shutterstock
both line and rotational symmetry
Practice
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 It is also possible to define a symmetry based
8.
rotational symmetry
Concepts and Skills
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Investigating Symmetry 876
In the space provided, sketch an example of a figure with the given characteristics. Possible answers are shown.
AVOID COMMON ERRORS Some students may stop when they have found one line of symmetry or one angle of rotation. Remind them to reread the directions to see if they are asked to find all lines of symmetry or all angles of rotation.
12. no line symmetry; angle of rotational symmetry: 180°
JOURNAL
13. one line of symmetry; no rotational symmetry
14. Describe the line and rotational symmetry in this figure.
Have students create four different, simple logos. For the first logo, there should be no rotations that map the logo onto itself. For the second, a rotation of 180° should map the logo onto itself; for the third, 120°; and for the fourth, 90°.
four lines of symmetry; angle of rotational symmetry: 90° H.O.T. Focus on Higher Order Thinking
15. Communicate Mathematical Ideas How is a rectangle similar to an ellipse? Use concepts of symmetry in your answer.
Both have two perpendicular lines of symmetry, and both have 180° rotational symmetry. 16. Explain the Error A student was asked to draw all of the lines of symmetry on each figure shown. Identify the student’s work as correct or incorrect. If incorrect, explain why.
© Houghton Mifflin Harcourt Publishing Company
a.
Incorrect; the two diagonals are not lines of symmetry. b.
Incorrect; the figure has no lines of symmetry. c.
Incorrect; the figure has three more lines of symmetry, each connecting the remaining pairs of opposite vertices.
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
9–11
2 Skills/Concepts
MP.2 Reasoning
12–13
2 Skills/Concepts
MP.4 Modeling
14
3 Strategic Thinking
MP.4 Modeling
15
2 Skills/Concepts
MP.6 Precision
16
3 Strategic/Thinking
MP.5 Using Tools
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Lesson Performance Task
AVOID COMMON ERRORS In evaluating rectangular shapes for symmetries, students sometimes identify the diagonals as lines of symmetry. Unless a rectangle is a square, its diagonals are not lines of symmetry.
FOCUS ON CRITICAL THINKING MP.3 Students should recognize that while single elements of their designs might exhibit symmetries, those symmetries might not extend to the entire design. For example, in a design composed of a square and four isosceles triangles, each of those shapes contains lines of symmetry. None of those lines, however, is a line of symmetry for the entire design.
Use symmetry to design a work of art. Begin by drawing one simple geometric figure, such as a triangle, square, or rectangle, on a piece of construction paper. Then add other lines or twodimensional shapes to the figure. Next, make identical copies of the figure, and then arrange them in a symmetric pattern. Evaluate the symmetry of the work of art you created. Rotate it to identify an angle of rotational symmetry. Compare the line symmetry of the original figure with the line symmetry of the finished work.
Answers will vary. Students’ responses should identify all lines of symmetry (horizontal, vertical, and diagaonal) as well as all angles of rotational symmetry (90°, 180°, and 270°). © Houghton Mifflin Harcourt Publishing Company
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The flags of many nations have rotational symmetry or line symmetry. The flags of a few nations, such as Jamaica, have both. Research the flags of the nations of the world to find examples of symmetry. If you wish, disregard color and concentrate only on the designs of the flags. Draw or print out examples of designs you find especially interesting or attractive.
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Investigating Symmetry 878
MODULE
17
STUDY GUIDE REVIEW
Transformations and Symmetry
Study Guide Review
Essential Question: How can you use transformations to solve real-world problems?
ASSESSMENT AND INTERVENTION
KEY EXAMPLE
(Lesson 17.1)
Translate the square ABCD along the vector ⧼2, 1⧽. A(1, 2), B(3, 2), C(1, 4), D(3, 4).
(x, y) → (x + a, y + b)
A(1, 2) → A'(1 + 2, 2 + 1)
Assign or customize module reviews.
B(3, 2) → B'(3 + 2, 2 + 1)
Write the rule for translation along the vector ⧼a, b⧽. Apply the rule to each point.
C(1, 4) → C'(1 + 2, 4 + 1) A'(3, 3), B'(5, 3),
KEY EXAMPLE
COMMON CORE
A(2, 3), B(3, 4), and C(3, 1) reflected across the line y = x.
(x, y) → (y, x) A(2, 3), A'(3, 2)
• What movements the bird made: Sample answer: translations up, rotations of 90° clockwise, translations down and to the right, translations to the right • How to represent the movements with coordinates: Make a table showing the coordinates after every transformation. • If some images include more than one transformation: Yes, some images can be a composition of a rotation and a translation, as is done with the second and third sets of coordinates in the sample answer. • How to perform the series of transformations: Sample answer: draw the new position of the figure, give its coordinates, and then describe the composition of transformations needed to find the image figure.
Module 17
© Houghton Mifflin Harcourt Publishing Company
SUPPORTING STUDENT REASONING Students should begin this problem by focusing on the transformations needed to simulate movement in the coordinate plane. Here are some issues they might bring up.
(Lesson 17.2)
Determine the vertices of the image of △ABC.
Mathematical Practices: MP.1, MP.4, MP.8 G-CO.A.2, G-CO.A.5, G-CO.B.6, G-MG.A.1
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Now simplify.
C'(3, 5), D'(5, 5)
B(3, 4), B'(4, 3)
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Key Vocabulary
vector (vector) initial point (punto inicial) terminal point (punto terminal) translation (translación) perpendicular lines (líneas perpendiculares) perpendicular bisector (mediatriz) reflection (reflexión) rotation (rotación) center of rotation (centro de rotación) angle of rotation (ángulo de rotación) symmetry (simetría)
D(3, 4) → D'(3 + 2, 4 + 1)
MODULE PERFORMANCE TASK
MODULE
Write the rule for reflection across the line y = x. Apply the rule to each point.
line symmetry (simetría de línea) line of symmetry (línea de simetría) rotational symmetry (simetría rotacional) angle of rotational symmetry (ángulo de simetría rotacional)
C(3, 1), C'(1, 3) KEY EXAMPLE
(Lesson 17.3)
Determine the vertices of the image of △DFE. D(1, 2), F(2, 2), and E(2, 0), rotated 270° counterclockwise about the origin.
(x, y) → (y, -x) D(1, 2), → D'(2, -1)
Write the rule for a rotation 270° counterclockwise. Apply the rule to each point.
F(2, 2), → F'(2, -2)
E(2, 0), → E'(0, -2)
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Study Guide Review
SCAFFOLDING SUPPORT
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• Students may find it useful to cut out a set of triangles congruent to the given triangle and place them on the graph to plan the flight of the bird. Students can use the triangles as patterns to trace the successive positions of the bird. • Explain that it is fine if the triangles overlap from one triangle to the next. • Encourage students to be creative in their flight plans. The general outline is “exit the perch, swoop down and right, then off to the right,” but students can add interesting movements to the basic plan if they wish.
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EXERCISES Translate each figure along each vector. (Lesson 17.1)
SAMPLE SOLUTION 1. translation (x, y) → (x, y + 2):
1. The line segment determined by A(4, 7) and B(2, 9) along ⧼0, -2⧽. A'(4, 5), B'(2, 7) . The vertices of the image are
(0, 6), (-2, 4), (2, 4) to (0, 8), (-2, 6), (2, 6)
2. The triangle determined by A(-3, 2), B(4, 4), and C(1, 1) along ⧼-1, -3⧽. A'(-4, -1), B'(3, 1), C'(0, -2) . The vertices of the image are
2. 90° clockwise rotation about (0, 6):
Determine the vertices of each image. (Lesson 17.2)
(0, 8), (-2, 6), (2, 6) to (0, 4), (0, 8), (2, 6)
3. The image of the rectangle ABCD reflected across the line y = -x.
A(-3, 2), B(3, 2), C(-3, -3), D(3, -3) A'(-2, 3), B'(-2, -3), C'(3, 3), D'(3, -3) . The vertices of the image are
3. translation (x, y) → (x + 4, y):
(0, 4), (0, 8), (2, 6) to (4, 4), (4, 8), (6, 6)
4. The image of the polygon ABCDE reflected across the x-axis.
A(-1, -1), B(0, 1), C(4, 2), D(6, 0), E(3, -3) A'(-1, 1), B'(0, -1), C'(4, -2), D'(6, 0), E'(3, 3) . The vertices of the image are
4. translation (x, y) → (x, y - 6):
Determine the vertices of the image. (Lesson 17.3)
(4, 4), (4, 8), (6, 6) to (4, -2), (4, 2), (6, 0)
5. The figure defined by A(3, 5), B(5, 3), C(2, 2) rotated 180° counterclockwise. A'(-3, -5), B'(-5, -3), C'(-2, -2) about the origin. The points of the image are
5. translation (x, y) → (x + 2, y):
(4, –2), (4, 2), (6, 0) to (6, –2), (6, 2), (8, 0)
MODULE PERFORMANCE TASK
Animating Digital Images A computer animator is designing an animation in which a bird flies off its perch, swoops down and to the right, and then flies off the right side of the screen. The graph shows the designer’s preliminary sketch, using a triangle to represent the bird in its initial position (top) and one intermediate position.
Module 17
880
6
2 x -2
0
2
6
© Houghton Mifflin Harcourt Publishing Company
Plan a series of rotations and translations to animate the flight of the bird. Sketch each rotation and translation on a graph and label the coordinates of the triangle’s vertices at each position. If you wish, you can test out how well your animation works by making a flipbook of your graphs.
y
Study Guide Review
DISCUSSION OPPORTUNITIES
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4/19/14 11:41 AM
• Can the transformations be done in the same way but explained differently? How can you verify this? • Why did you need to represent the triangle’s movement as a series of individual transformations, instead of a single composition of transformations? One transformation would not work for animation, because intermediate positions need to be shown.
Assessment Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem.
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Ready to Go On?
Ready to Go On?
17.1–17.4 Transformations and Symmetry
ASSESS MASTERY Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
• Online Homework • Hints and Help • Extra Practice
_ _ 1. Line segment YZ was used to translate ABCDE. YZ is 6.2 inches long. What is the length of AA' + BB' + CC' + DD' + EE' ? (Lesson 17.1) 31 inches
ASSESSMENT AND INTERVENTION
Given figure FGHI and its image F'G'H'I', answer the following. (Lesson 17.2, 17.3) 2a. Write an algebraic rule for the rotation shown and then describe the rotation in words.
F(-10, 6) → F'(10, -6), G(-6, 9) → G'(6, -9),
H(-3, 5) → H'(3, -5), I(-7, 2) → I'(7, -2), The
is a rotation of 180° . 2b. Tell whether the figure FGHI has line symmetry, rotational symmetry, both types of symmetry, or no symmetry. If the figure has line symmetry, record the number. If the figure has rotational symmetry, list the angles of rotation that are less than 360°.
ADDITIONAL RESOURCES
Differentiated Instruction Resources • Reading Strategies • Success for English Learners • Challenge Worksheets Assessment Resources
© Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
Types of symmetry
Number of lines of symmetry
line, rotational
4
y
8
F
rule is (x, y) → (-x, -y), and the transformation
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Response to Intervention Resources
G H 4
x
I -8
-4
0 -4
H′
-8
G′
Angles of rotation 90°, 180°, 270°
y=x+2
Essential Question 4. In which situations are translations useful for transformations? Reflections? Rotations?
Answers may vary. Sample: Translations are useful when shifting the points of an image along the same direction and length, reflections are useful when mirroring an image across a line, and rotations are useful when shifting an image around a fixed point.
Module 17
COMMON CORE
Study Guide Review
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Common Core Standards
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Module 17
F′
3. Given triangle ABC with A(-2, 4), B(-2, 1), and C(-4, 0), and its image A'B'C' with A'(2, 0), B'(-1, 0), and C'(-2, -2), find the line of reflection. (Lesson 17.2)
• Leveled Module Quizzes
881
I′8
4
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Content Standards Mathematical Practices
Lesson
Items
16.1
1
G-CO.A.4
MP.4
16.3, 16.4
2
G-CO.A.4, G-CO.A.2, G-CO.A.5, G-CO.A.6
MP.7
16.2
3
G-CO.A.4, G-CO.A.5, G-CO.A.6
MP.4
MODULE MODULE 17 MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. Triangle ABC is given by the points A(-1, 5), B(0, 3), and C(2, 4). It is reflected over the line y = -2x - 2. Does the image contain each of the points? Select Yes or No for A–C. A. A' (-5, 3x)
Yes Yes Yes
B. B' (-4, 6) C. C' (-6, 0)
17
ASSESSMENT AND INTERVENTION
No No No
2. A triangle, △ABC, is rotated 90° counterclockwise, reflected across the x-axis, and then reflected across the y-axis. Choose True or False for each statement. A. Rotating △ABC 180° clockwise is an True False equivalent transformation. B. Rotating △ABC 270° counterclockwise is an equivalent transformation. C. Reflecting △ABC across the y-axis is an equivalent transformation.
True
False
True
False
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
3. Choose True or False for each statement about equilateral triangles. A. An equilateral triangle has 3 equal angle True False measures. B. An equilateral triangle has 3 equal side True False measures. C. An equilateral triangle has 3 lines of True False symmetry.
ADDITIONAL RESOURCES Assessment Resources • Leveled Module Quizzes: Modified, B
Answers may vary. Sample: Because when the figure is folded along the lines, one half coincides with the other. The angles of rotational symmetry are 72°, 144°, 216°, and 288°.
Module 17
COMMON CORE
AVOID COMMON ERRORS Item 4 Some students will follow the rule for reflecting over the x- or y-axis when trying to reflect over the line y = x. Remind students that they can graph the points and the line y = x on graph paper, and then fold the paper along the line of symmetry to see where the points will land.
© Houghton Mifflin Harcourt Publishing Company
4. A line segment with points P(1, 2) and Q(4, 3) is reflected across the line y = x. What are the new coordinates of the points of the line segment? P' (2, 1) , Q' (3, 4) 5. Draw on the figure all lines of symmetry and explain why those lines are the lines of symmetry. Give all angles of rotational symmetry less than 360°.
Study Guide Review
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Common Core Standards
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Content Standards Mathematical Practices
Lesson
Items
16.2
1
G-CO.A.3
MP.5
16.2, 16.3, 15.3
2*
G-CO.B.6
MP.5
16.4, 15.2
3*
G-CO.A.5, G-CO.A.1
MP.5
16.2
4
G-CO.B.6
MP.7
16.4
5
G-CO.A.5
MP.6
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 882
MODULE
18
Congruent Figures
Congruent Figures ESSENTIAL QUESTION: Answer: The principles of congruency can be used to establish whether two objects in the real world are the same shape or not.
Essential Question: How can you use congruency
to solve real-world problems?
18 MODULE
LESSON 18.1
Sequences of Transformations LESSON 18.2
Proving Figures Are Congruent Using Rigid Motions
This version is for
Algebra 1 and PROFESSIONAL DEVELOPMENT Geometry only VIDEO
LESSON 18.3
Corresponding Parts of Congruent Figures Are Congruent
Professional Development Video
Professional Development my.hrw.com
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (t) ©Scott E. Feuer/Shutterstock; (b) ©Hitdelight/Shutterstock
Author Juli Dixon models successful teaching practices in an actual high-school classroom.
REAL WORLD VIDEO Check out how landscape architects use transformations of geometric shapes to design green space for parks and homes.
MODULE PERFORMANCE TASK PREVIEW
Jigsaw Puzzle In this module, you will use congruency and a series of transformations to solve a portion of a jigsaw puzzle. What is some of the basic geometry behind a jigsaw puzzle? Let’s get started on finding out how all the pieces fit together!
Module 18
DIGITAL TEACHER EDITION IN1_MNLESE389762_U7M18MO 883
Access a full suite of teaching resources when and where you need them: • Access content online or offline • Customize lessons to share with your class • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most
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Module 18
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PERSONAL MATH TRAINER Assessment and Intervention Assign automatically graded homework, quizzes, tests, and intervention activities. Prepare your students with updated, Common Core-aligned practice tests.
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Are YOU Ready?
Are You Ready?
Complete these exercises to review skills you will need for this module.
ASSESS READINESS
Properties of Reflections Example 1
Find the points that define the reflection of the figure given by A(1, 1), B(2, 3), and C(3, 1) across the y-axis.
4
B
0
-2
-2 Use the rules for reflections on a coordinate plane. For a -4 reflection across the y-axis: (x, y) → (–x, y) A(1, 1) → A′(–1, 1), B(2, 3) → B′(–2, 3), C(3, 1) → C′(–3, 1)
C 2
Use the assessment on this page to determine if students need strategic or intensive intervention for the module’s prerequisite skills.
• Online Homework • Hints and Help • Extra Practice
B
2 A A
C -4
y
x 4
ASSESSMENT AND INTERVENTION
Find the vertices of the reflected figure. 1.
△ABC reflected across the x-axis A′ (1, –1), B′ (2, –3), C′ (3, –1)
2.
△ABC reflected across y = x
3 2
A′ (1, 1), B′ (3, 2), C′ (1, 3)
1
Personal Math Trainer will automatically create a standards-based, personalized intervention assignment for your students, targeting each student’s individual needs!
Properties of Rotations Example 2
Find the vertices of △ABC rotated 90° counterclockwise around the origin. (x, y) → (–y, x)
Write the rule for rotation.
C(3, 1) → C′(–1, 3)
Apply the rule.
© Houghton Mifflin Harcourt Publishing Company
A(1, 1) → A′(–1, 1), B(2, 3) → B′(–3, 2),
Find the vertices of the rotated figure. 3.
△ABC rotated 180° around the origin
A′ (–1, –1), B′ (–2, –3), C′ (–3, –1)
Properties of Translations Example 3
Calculate the vertices of the image of △ABC translated using the rule (x, y) → (x + 2, y + 1). A(1, 1) → A′(3, 2), B(2, 3) → B′(4, 4), C(3, 1) → C′(5, 2)
TIER 1, TIER 2, TIER 3 SKILLS
ADDITIONAL RESOURCES See the table below for a full list of intervention resources available for this module. Response to Intervention Resources also includes: • Tier 2 Skill Pre-Tests for each Module • Tier 2 Skill Post-Tests for each skill
Apply the rule.
Calculate the vertices of the image. 4.
△ABC translated using the rule (x, y) → (x – 2, y + 2)
Module 18
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Tier 1 Lesson Intervention Worksheets Reteach 18.1 Reteach 18.2 Reteach 18.3
A′(–1, 3), B′(0, 5), C′(1, 3)
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Response to Intervention Tier 2 Strategic Intervention Skills Intervention Worksheets
Tier 3 Intensive Intervention Worksheets available online
37 Congruent Figures 42 Properties of Reflections 43 Properties of Rotations 44 Properties of Translations
Building Block Skills 8, 16, 46, 48, 53, 56, 74, 98, 102, 103
Differentiated Instruction
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Challenge worksheets Extend the Math Lesson Activities in TE
Module 18
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LESSON
18.1
Name
Sequences of Transformations
Essential Question: What happens when you apply more than one transformation to a figure?
Resource Locker
Explore
The student is expected to:
Find the image for each sequence of transformations.
Mathematical Practices
MP.5 Using Tools
Select tools, including real objects, manipulatives, paper and pencil, Explain to a partner why a transformation or sequence of transformations is rigid or nonrigid.
A′
B″
B′ C″
A B
C′ C
Make a conjecture regarding a single rotation that will map △ABC to △A″B″C″. Check your conjecture, and describe what you did.
A rotation of 75° (because 30 + 45 = 75) should map △ABC to △A″B″C″. By using the software to rotate △ABC 75°, I can see that this image coincides with △A″B″C″.
© Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON PERFORMANCE TASK
A″
P
ENGAGE
Possible answer: The transformations occur sequentially, and order matters. The result may be the same as a single transformation.
Using geometry software, draw a triangle and label the vertices A, B, and C. Then draw a point outside the triangle and label it P. Rotate △ABC 30° around point P and label the image as △A′B′C ′. Then rotate △A′B′C ′ 45° around point P and label the image as △A″B″C ″. Sketch your result.
Language Objective
Essential Question: What happens when you apply more than one transformation to a figure?
Combining Rotations or Reflections
A transformation is a function that takes points on the plane and maps them to other points on the plane. Transformations can be applied one after the other in a sequence where you use the image of the first transformation as the preimage for the next transformation.
G-CO.A.5
... Specify a sequence of transformations that will carry a given figure onto another. Also G-CO.A.2, G-CO.B.6 COMMON CORE
Date
18.1 Sequences of Transformations
Common Core Math Standards COMMON CORE
Class
Using geometry software, draw a triangle and label the vertices D, E, and F. Then draw two lines and label them j and k.
D
j
E′
D′
k D″
Reflect △DEF across line j and label the image as △D′E′F ′. Then reflect △D′E′F ′ across line k and label the image as △D″E″F ″. Sketch your result.
View the Engage section online. Discuss the photo and ask students to describe the snowflake in general terms, such as “It has six arms that look alike.” Then preview the Lesson Performance Task.
E
F
E″
F′ F″
Consider the relationship between △DEF and △D″E″F″. Describe the single t ransformation that maps △DEF to △D″E″F″. How can you check that you are correct?
A rotation with center at the intersection of j and k maps △DEF to △D″E″F″. Rotating △DEF around the intersection of j and k by the angle made between the lines rotates it about halfway to △D″E″F″, so rotate it by twice that angle to see △DEF mapped to △D″E″F″. Module 18
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A′
C″
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HARDCOVER PAGES 885896
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Lesson 18.1
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Reflect
1.
EXPLORE
Repeat Step A using other angle measures. Make a conjecture about what single transformation will describe a sequence of two rotations about the same center. If a figure is rotated and then the image is rotated about the same center, a single rotation
Combining Reflections
by the sum of the angles of rotation will have the same result. 2.
INTEGRATE TECHNOLOGY
Make a conjecture about what single transformation will describe a sequence of three rotations about the same center. A sequence of three rotations about the same center can be described by a single rotation
Students have the option of completing the combining reflections activity either in the book or online.
by the sum of the angles of rotation. 3.
Discussion Repeat Step C, but make lines j and k parallel instead of intersecting. Make a conjecture about what single transformation will now map △DEF to △D″E″F ″. Check your conjecture and describe what you did. △D″E″F″ looks like a translation of △DEF. I marked a vector from D to D″ and
QUESTIONING STRATEGIES
translated △DEF by it . The image coincides with △D″E″F″, so two reflections in || lines
How can you use geometry software to check your transformations? For reflections in parallel lines, use the measuring features to see if all points move the same distance in the same direction. For reflections in intersecting lines, rotate the preimage figure to see if the images are the same size and shape.
result in a translation.
Explain 1
Combining Rigid Transformations
In the Explore, you saw that sometimes you can use a single transformation to describe the result of applying a sequence of two transformations. Now you will apply sequences of rigid transformations that cannot be described by a single transformation. Example 1
Draw the image of △ABC after the given combination of transformations.
Reflection over line ℓ then translation along ⇀ v
⇀ ν
EXPLAIN 1 © Houghton Mifflin Harcourt Publishing Company
ℓ A C B
Step 1 Draw the image of △ABC after a reflection across line ℓ. Label the image △A′B′C ′.
v. Step 2 Translate △A′B′C ′ along ⇀ Label this image △A″B″C ″.
⇀ ν
⇀ ν C′
B′ A′ A
ℓ
Some students may transform the original figure twice instead of transforming the first image to get the second, and the second to get the third. Note that when performing two transformations with A → A' as the first transformation, A is the preimage and A' is the image. In the second transformation A' → A", A' is the preimage and A" is the image.
A″
A′ A
B
AVOID COMMON ERRORS
ℓ
B′
C
Module 18
C″
C′ B″
Combining Rigid Transformations
C B
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Lesson 1
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M18L1 886
Math Background
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Students have worked with individual transformations and should now be able to identify and describe translations, reflections, and rotations. In this lesson, they combine two or more of these transformations and may include sequences of nonrigid transformations. They must be able to visualize and predict the outcome of performing more than one transformation, as well as consider other transformations that produce the same final image. Throughout the lesson they must recall the properties of each transformation and the methods for drawing them.
Sequences of Transformations
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180° rotation around point P, then translation along ⇀ v , then reflection across line ℓ
B
QUESTIONING STRATEGIES After a rigid motion, an image has the same shape and size as the preimage. If you perform a sequence of rigid motions, will the final image have the same shape and size as the original? Yes; each rigid motion preserves size and shape, so a sequence of rigid motions will also preserve size and shape.
B′′ B′
Apply the rotation. Label the image △A′B′C ′.
A′
Apply the translation to △A′B′C ′. Label the image △A″B″C ″.
C
A′′ C′′ ℓ C″′
P C′
Apply the reflection to △A″B″C ″. Label the image △A‴B‴C ‴.
A B
ν
⇀
A″′ B″′
Reflect
4.
Are the images you drew for each example the same size and shape as the given preimage? In what ways do rigid transformations change the preimage? Yes. Rigid transformations move the figure in the plane and may change the orientation,
but they do not change the size or shape.
5.
Does the order in which you apply the transformations make a difference? Test your conjecture by performing the transformations in Part B in a different order. Possible answer: Yes, if I reflect first, then rotate, and then translate, the final image is
above line ℓ instead of below it.
6.
For Part B, describe a sequence of transformations that will take △A″B″C″ back to the preimage. Possible answer: In this case, reversing the order of the transformations will take the final
image back to the preimage.
© Houghton Mifflin Harcourt Publishing Company
Your Turn
Draw the image of △ABC after the given combination of transformations. 7.
Reflection across ℓ then 90° rotation around point P A′′ B′′
B
8.
Translation along ⇀ v then 180° rotation around point P then translation along ⇀ u E′
ℓ B′
E G′ G
C′′ A
C P
C′
F′′′
A′
Module 18
F′ F′′
F P
G′′′ E′′′
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E′′
⇀
u
G′′ ⇀ v
Lesson 1
COLLABORATIVE LEARNING IN1_MNLESE389762_U7M18L1 887
Small Group Activity Geometry software allows students to focus on their predictions rather than on drawing multiple transformations. Give students the coordinates of a figure and a series of transformations. Instruct them to plot the points on graph paper and to sketch a prediction of the final image. Then have them use geometry software to perform the transformations and check the results against their predictions. After students have done this for several figures, ask them to brainstorm ways to make their predictions more accurate.
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Lesson 18.1
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Explain 2 Example 2
Combining Nonrigid Transformations
(
)
Combining Nonrigid Transformations
3 x, _ 3 y → -x, y → x + 1, y - 2 (x, y) → _ ( ) ( ) 2 2 1. The first transformation is a dilation by a factor of __32 . Apply the dilation. Label the image A′B′C′D′.
y
B′′
D′′
A′′ D′′′
-8
(
-6
A′′′
-4 -2
C′
B
4 2
C A′
A
0
2
D′
D 4
6
6
Apply the stretch. Label the image △A′B′C′ .
C′′
2. The second transformation is a dilation by a factor combined with a reflection. Apply the transformation to △A′B′C′ . Label the
8
y
QUESTIONING STRATEGIES
4 2
3
_1 2
B′′
-6
-4 -2
image △A″B″C″.
C′
B′
C
B
0 A′
2 A
-4
Reflect
If you dilated a figure by a factor of 2, what transformation could you use to return the figure back to its preimage? If you dilated a figure by a factor of 2 and then translated it right 2 units, write a sequence of transformations to return the figure back to its preimage. 1 . Possible answer: You could dilate the figure by a factor Dilate the figure by a factor of _ 2 1 of _ then translate the figure left 2 units. 2
10. A student is asked to reflect a figure across the y-axis and then vertically stretch the figure by a factor of 2. Describe the effect on the coordinates. Then write one transformation using coordinate notation that combines these two transformations into one. The x-coordinates change to their opposites. The y-coordinates are multiplied by a factor
© Houghton Mifflin Harcourt Publishing Company
-6
9.
How would you describe the image of a figure after a sequence of nonrigid transformations? Either the size or the shape of the original figure changed, although it is possible that a subsequent transformation results in a figure of the original size and shape.
x
A′′ -12 -10 -8
transformation by comparing the size and shape of the original and image figures. Point out that a dilation preserves the shape but not the size of a figure, while a horizontal or vertical stretch does not preserve either the size or the shape of a figure.
x
)
1. The first transformation is a [horizontal/vertical] stretch by a factor of 3 .
of
B′
6 B′′′
C′′′
3. Apply the translation of A″B″C ″D″. Label this image A″'B″'C ″'D″'.
1 x, -_ 1y (x, y) → (3x, y) → _ 2 2
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Relate nonrigid transformation to rigid
8 C′′
2. Apply the reflection of A′B′C ′D ′ across the y-axis. Label this image A″B″C ″D″.
EXPLAIN 2
Draw the image of the figure in the plane after the given combination of transformations.
If you perform a sequence of nonrigid motions on a polygon, will the type of polygon change? Explain. No. The polygon will have the same number of vertices, so it will be the same general polygon. If the original figure is regular, the nonrigid motions may give an image of a non-regular polygon. The image of a square may be a parallelogram, for example.
of 2. (x, y) → (-x, 2y)
CONNECT VOCABULARY Module 18
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Multiple Representations Have students graph any three points on a coordinate plane and connect them to form a triangle. Ask students to perform two transformations on this triangle. Then instruct them to use the algebra rules to perform the same transformations. Students should compare the coordinates they found algebraically with those they found with the physical transformation. Then have them study the preimage and the final image to decide whether they could have used one transformation to obtain the same result. If so, ask them to use the algebraic rules to show that the single transformation is equivalent to the two original transformations.
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The word rigid derives from rigidus, the Latin word for stiff. Help students understand how nonrigid transformation is used to represent a type of transformation that gives an image that is a different size and/or shape of a preimage figure. Point out that the transformation can be in the plane or in the coordinate plane, and that a nonrigid transformation can be included in any sequence of combined transformations.
Sequences of Transformations
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Your Turn
EXPLAIN 3
Draw the image of the figure in the plane after the given combination of transformations. 11. (x, y) → (x - 1, y - 1) → (3x, y) → (-x, -y)
Predicting the Effect of Transformations
6 4
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Encourage students to predict the effect of
2 -6 C′′′
-4 -2 A′′′
B′′′
transformations and then actually perform the transformations described in the example to verify their predictions. Have students repeat the same sequence of transformations using a different figure as the original figure. Ask whether the sequence of transformations affects the new figure in the same way.
Explain 3 Example 3
0 -2
(
y B′
B
6 B′′
A C A′ C′ A′′ 2
C′′ x 4
2
C
B
x
A
B′′ -6
6
-4 -2
0
2
-2
-4
-4
-6
-6
C′′
A′
4
6
B′ C′
Predicting the Effect of Transformations
Predict the result of applying the sequence of transformations to the given figure.
△LMN is translated along the vector 〈-2, 3〉, reflected across the y-axis, and then reflected across the x-axis.
6 L
-6
Predict the effect of the first transformation: A translation along the vector 〈-2, 3〉 will move the figure left 2 units and up 3 units. Since the given triangle is in Quadrant II, the translation will move it further from the x- and y-axes. It will remain in Quadrant II.
y
4 N
M
© Houghton Mifflin Harcourt Publishing Company
y
4 A′′
QUESTIONING STRATEGIES Why is it important to carefully label the vertices after each transformation? Labeling the vertices will help distinguish the types of rigid and nonrigid transformations used in the sequence of transformations. Mislabeling a transformation in the sequence will likely result in an incorrect final image.
)
3 x, -2y → x - 5, y + 4 12. (x, y) → _ ( ) 2
-4
x -2
0
2
4
6
-2 -4 -6
Predict the effect of the second transformation: Since the triangle is in Quadrant II, a reflection across the y-axis will change the orientation and move the triangle into Quadrant I. Predict the effect of the third transformation: A reflection across the x-axis will again change the orientation and move the triangle into Quadrant IV. The two reflections are the equivalent of rotating the figure 180° about the origin. The final result will be a triangle the same shape and size as △LMN in Quadrant IV. It has been rotated 180° about the origin and is farther from the axes than the preimage. Module 18
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Communicate Math Have students work in pairs. Have the first student show the partner a graph of a preimage and transformed image and ask whether it is an example of a rigid or nonrigid transformation. The second student should describe the transformation and tell whether it is rigid or nonrigid, and why. The first student writes the explanation under the images. Students change roles and repeat the sequence with another set of images.
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B
Square HIJK is rotated 90° clockwise about the origin and then dilated by a factor of 2, which maps (x, y) → (2x, 2y).
6
Predict the effect of the first transformation: A 90° clockwise
4
rotation will map it to Quadrant IV. Due to its
2
symmetry, it will appear to have been translated, but will be closer to the x-axis than it is to the y-axis.
-6
-4
-2
y H
I
K
J
0
2
x 4
6
-2
Predict the effect of the second transformation: A dilation
-4
by a factor of 2 will double the side lengths of the
-6
square. It will also be further from the origin than the preimage. The final result will be a square in Quadrant 4 with side lengths twice as long as the
side lengths of the original. The image is further from the origin than the preimage.
Your Turn
Predict the result of applying the sequence of transformations to the given figure. 13. Rectangle GHJK is reflected across the y-axis and translated along the vector 〈5, 4〉.
4
(
K
J
2 x -6
-4
-2
0
6
-2 -4
G
H
-6
14. △TUV is horizontally stretched by a factor of __32 , which maps (x, y) → __32 x, y , and then translated along the vector 〈2, 1〉.
)
A horizontal stretch will pull points U and T away from the y-axis, making the triangle longer in the leftto-right direction. The translation along the vector 〈2, 1〉 will move the stretched triangle 2 units right and 1 unit up, which will move the triangle closer to the origin with one vertex on the x-axis and another across the y-axis. The final image will not be the same shape or size as the preimage.
y
6
y
4 2 x -6
-4 T
U 0
2
4
6
© Houghton Mifflin Harcourt Publishing Company
The reflection across the y-axis will move the rectangle from the right of the y-axis to the left of it. Due to the symmetry of the rectangle, it will appear to have been translated left 6 units. Then, translating along the vector 〈5, 4〉 will move the rectangle right 5 units and up 4 units. This will bring the rectangle fully into Quadrant I. The final result will be a rectangle that is the same shape and size as the preimage that has moved to sit on the x-axis in Quadrant I, closer to the y-axis than the preimage.
6
-4 V -6
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Sequences of Transformations
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Elaborate
ELABORATE
15. Discussion How many different sequences of rigid transformations do you think you can find to take a preimage back onto itself? Explain your reasoning. An infinite number. With rotations you just need to go 360° and you will be back where
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Discuss with students how to record and use
you started, and you can do that as many times as you want. You can always reflect back over a line. You can always go back left just as far as you went right, or up as many times as you went down in a translation, so you can take a preimage back onto itself in many ways. You can add extra transformations to find additional sequences.
algebraic patterns to represent a series of rigid and nonrigid motions in the coordinate plane. For example, a reflection in the x-axis is represented by (x, y) → (x, –y), while a reflection in the y-axis is represented by (x, y) → (–x, y). So, a reflection in the x-axis followed by a reflection in the y-axis is represented by (x, y) → (–x, –y).
16. Is there a sequence of a rotation and a dilation that will result in an image that is the same size and position as the preimage? Explain your reasoning. Yes, a rotation of 360° and a dilation of 1 will work.
QUESTIONING STRATEGIES Is a transformation from a sequence of rigid motions always rigid? Yes. Each rigid motion preserves the size and shape of a figure, so the final image must have the same size and shape as the original figure.
added together to make one rotation, even if they are a combination of clockwise and © Houghton Mifflin Harcourt Publishing Company
What types of sequences of transformations can be undone? A sequence of transformations using the same rigid motion can sometimes be undone by reversing the order of the sequence. It is possible that a sequence using different rigid motions or nonrigid motions cannot be undone directly, but students may be able to write a series of related transformations to undo the transformations.
17. Essential Question Check-In In a sequence of transformations, the order of the transformations can affect the final image. Describe a sequence of transformations where the order does not matter. Describe a sequence of transformations where the order does matter. Possible answer: A sequence of any number of rotations about the same point can be
counterclockwise rotations. Any sequence of translations can also be done in any order. When a sequence includes a mix of different types of transformations, the order usually affects the final image, for example a rotation of 90° around a vertex followed by a dilation by a factor of 4 will have a different final image than the same figure dilated by a factor of 4 followed by a rotation of 90°.
SUMMARIZE THE LESSON What features can you describe when predicting the result of more than one transformation? Sample answer: You can predict which quadrant(s) the final image will be in, how far and in what direction it will be from the origin or from the original figure, its orientation, and whether the size or shape of the figure has changed.
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Evaluate: Homework and Practice
EVALUATE • Online Homework • Hints and Help • Extra Practice
Draw and label the final image of △ABC after the given sequence of transformations. 1.
Reflect △ABC over the y-axis and then translate by 〈2, -3〉. 6 C B
3.
-4 -2
-2
C
C′ B′ C′′ A′ 2
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B
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6
A′′
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-6
-6
2
6
8
-8
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B′
B′′ C′′
A′′′
2
x 0
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Example 2 Combining Nonrigid Transformations
Exercises 5–6
Example 3 Predicting the Effect of Transformations
Exercises 7–12
2
4
6
8
sequence of transformations or to check a sequence of transformations. Remind students to use the measuring features to verify that a sequence of rigid motions preserves the size and shape of a figure.
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Depth of Knowledge (D.O.K.)
B′′′
4
-4 -2 A′′
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COMMON CORE
Mathematical Practices
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2 Skills/Concepts
MP.6 Precision
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2 Skills/Concepts
MP.2 Reasoning
1 Recall of Information
MP.2 Reasoning
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2 Skills/Concepts
MP.2 Reasoning
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3 Strategic Thinking
MP.2 Reasoning
2 Skills/Concepts
MP.2 Reasoning
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Exercises 1–4
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can use geometry software to do a
x 0
C
Example 1 Combining Rigid Transformations
B′′′
2
-4 -2
Reflect △ABC over the x-axis, translate by 〈-3, -1〉, and rotate 180 degrees around the origin.
C′′′
© Houghton Mifflin Harcourt Publishing Company
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A′′′
A′
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B A
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C′
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C
Practice
Explore Combining Reflections
4 6 C′′ B′′
y
B′′
Exercise
Concepts and Skills
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8 B′ A′′
Module 18
C′
A A′ -4 -2 0 A′′ 2 -2
Translate △ABC by 〈4, 4〉, rotate 90 degrees counterclockwise around A, and reflect over the y-axis.
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ASSIGNMENT GUIDE
B′
2 B′′ x
C′′
4.
y
6
2 -6
Rotate △ABC 90 degrees clockwise about the origin and then reflect over the x-axis.
y
4
A 0
2.
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Draw and label the final image of △ABC after the given sequence of transformations.
GRAPHIC ORGANIZERS
5.
Suggest that students use a graphic organizer to keep track of the types of transformations and their properties in a sequence of transformations. This can help them remember to use the last image figure as they proceed with the sequence of transformations. For example:
( )
1 y → -2x, -2y ( ) (x, y) → x, _ 3 B 8
-6
Transformation 2 Property 1 Property 2
C′′′
C′
C′
-6A′ -4 -2
-8
4
C′′
Transformation 3 Property 1 Property 2
B′′
8 6
8.
B′ A
© Houghton Mifflin Harcourt Publishing Company
-6 B′′′
x
-8
6
Exercise
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-4 -2
B′′
B′
x
0
4C
A′′ 2 -2 A′′′ -4
6
C′
-6
B′′
B′′′
Possible answer: The translation moves the figure down one unit and left three units, mapping A’ to the left of the y-axis and C’ closer to the origin. The reflection first will map A’’B’’C’’ below the x-axis and change the orientation. The second reflection will map the figure mostly into Quadrant III, with A’’’ in Quadrant IV, and again change the orientation. The final image is the same size, shape, and orientation as the preimage.
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-4 -2 C′′′ 0 2 C′′ 4 -2 A′′ A′′′ -4
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Lesson 18.1
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B′′′
A′′
y C′′′
B
C′
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A
2
△ABC is translated along the vector 〈-1, -3〉, rotated 180° about the origin, and then dilated by a factor of 2.
y
2
Have students work in small groups to make a poster showing how to find a sequence of transformations in the coordinate plane using both rigid and nonrigid motions. Give each group a different sequence to transform. Then have each group present its poster to the rest of the class, explaining each step.
0
-6
A′ 4
SMALL GROUP ACTIVITY
B′′ x
-2 -4
△ABC is translated along the vector 〈-3, -1〉, reflected across the x-axis, and then reflected across the y-axis.
↓
Some students may perform a combination of transformations in the wrong order. Emphasize the importance of doing the transformations in the correct order by asking them to rotate a triangle 90° in the plane and then reflect it in the x-axis. They will get a different result if the order is reversed.
2
Predict the result of applying the sequence of transformations to the given figure. 7.
AVOID COMMON ERRORS
C
x
A′ 2
)
y
4
A
-4
↓
(
A′′′ B′ 6
C
-4 -2 0 A′′ -2
C′′
)
B 8
4 B′ 2 -6
(
3 x, _ 3y 2 y → x + 6, y - 4 → _ 2 x, -_ (x, y) → -_ ( ) 2 3 3 2
y
6
Transformation 1 Property 1 Property 2
6.
Possible answer: The translation moves the figure down and to the left without changing the shape or orientation. The rotation about the origin moves the figure from Quadrants I and IV to Quadrants II and III without changing the orientation. The dilation doubles the side lengths. The final image is the same shape as the preimage but larger. It has the same orientation. Lesson 1
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Depth of Knowledge (D.O.K.)
-8
COMMON CORE
Mathematical Practices
3 Strategic Thinking
MP.6 Precision
2 Skills/Concepts
MP.2 Reasoning
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In Exercises 9–12, use the diagram. Fill in the blank with the letter of the correct image described. G
B is the result of the sequence: G reflected over a vertical line and then a horizontal line. 10. E is the result of the sequence: D rotated 90° clockwise around one of its vertices and then reflected over a horizontal line. 11. F is the result of the sequence: E translated and then rotated 90° 9.
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 When writing the algebraic rules for the rigid
A
B F
counterclockwise. 12. A is the result of the sequence: D rotated 90° counterclockwise and then translated.
E
motions and other transformations, review the quadrants and coordinates. Students should remember that a rotation through a positive angle is in the counterclockwise direction, and a rotation through a negative angle is in the clockwise direction.
C D
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Have students use the algebraic representation
Choose the correct word to complete a true statement. 13. A combination of two rigid transformations on a preimage will always/sometimes/never produce the same image when taken in a different order.
14. A double rotation can always/sometimes/never be written as a single rotation.
15. A sequence of a translation and a reflection always/sometimes/never has a point that does not change position.
16. A sequence of a reflection across the x-axis and then a reflection across the y-axis always/sometimes/never results in a 180° rotation of the preimage.
17. A sequence of rigid transformations will always/sometimes/never result in an image that is the same size and orientation as the preimage.
18. A sequence of a rotation and a dilation will always/sometimes/never result in an image that is the same size and orientation as the preimage.
a. Reflect across the y-axis and then translate along the vector 〈0, -4〉.
Yes
No
b. Translate along the vector 〈0, -4〉 and then reflect across the y-axis.
Yes
No
c. Rotate 90° clockwise about the origin, reflect across the x-axis, and then rotate 90° counterclockwise about the origin.
Yes
No
d. Rotate 180° about the origin, reflect across the x-axis, and then translate along the vector 〈0, 4〉.
Yes
No
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y
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R 0 -2
x 2
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19. △QRS is the image of △LMN under a sequence of transformations. Can each of the following sequences be used to create the image, △QRS, from the preimage, △LMN? Select yes or no.
of a dilation in the coordinate plane as (x, y) → (kx, ky), where k is the scale factor. Ask students how they would represent the dilation that would “undo” this 1y 1 x, __ dilation. (x, y) → __ k k
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Sequences of Transformations
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y
20. A teacher gave students this puzzle: “I had a triangle with vertex A at (1, 4) and vertex B at (3, 2). After two rigid transformations, I had the image shown. Describe and show a sequence of transformations that will give this image from the preimage.”
PEERTOPEER DISCUSSION Ask students to discuss with a partner how to predict the final image for a combination of rigid transformations and nonrigid transformations. Then ask students to make a conjecture about the result of dilating a triangle whose vertices have coordinates (1, 1), (1, 4), (5, 1) with a scale factor of 2, followed by a reflection in the x-axis. The image has vertices (2, –2), (2, –8), (10, –2).
Possible answer: Translate by the vector 〈2,1〉 then reflect over the line x = 5.
Translation A′ A′′
6 4
A
Preimage 2
B 2C
0
B′ B′′ Reflection C′′ x 6
C′ 4
H.O.T. Focus on Higher Order Thinking
y
21. Analyze Relationships What two transformations would you apply to △ABC to get △DEF? How could you express these transformations with a single mapping rule in the form of (x, y) → (?, ?)?
2
Possible answer: Reflect △ABC across the y-axis and then translate it down 7 units. A single mapping rule would be (x, y) → (-x, y - 7).
JOURNAL Have students compare and contrast the methods they have learned for combining rigid transformations and nonrigid transformations in the coordinate plane.
A
4
-4
-2 D
0
C
B 2
x 4
-2 -4
F
E
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©AJP/ Shutterstock
22. Multi-Step Muralists will often make a scale drawing of an art piece before creating the large finished version. A muralist has sketched an art piece on a sheet of paper that is 3 feet by 4 feet. a. If the final mural will be 39 feet by 52 feet, what is the scale factor for this dilation?
Scale factor: 13
b. The owner of the wall has decided to only give permission to paint on the lower half of the wall. Can the muralist simply use the transformation (x, y) → x, __12 y in addition to the scale factor to alter the sketch for use in the allowed space? Explain. Only if the artist wants the final version of the mural to be distorted. This mapping will shrink the height of the mural in half, but by keeping the original width, the shapes will change.
(
)
23. Communicate Mathematical Ideas As a graded class activity, your teacher asks your class to reflect a triangle across the y-axis and then across the x-axis. Your classmate gets upset because he reversed the order of these reflections and thinks he will have to start over. What can you say to your classmate to help him?
The order of these two reflections does not matter. The resulting image is the same for a reflection in the y-axis followed by a reflection in the x-axis as for a reflection in the x-axis followed by a reflection in the y-axis.
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 A visual pattern can be described as a form or
The photograph shows an actual snowflake. Draw a detailed sketch of the “arm” of the snowflake located at the top left of the photo (10:00 on a clock face). Describe in as much detail as you can any translations, reflections, or rotations that you see.
shape that repeats. Ask students to describe the snowflake in terms of patterns. Sample answer: Each half of any one of the arms can be taken as a pattern that repeats approximately twelve times in the snowflake design, twice on each arm.
Then describe how the entire snowflake is constructed, based on what you found in the design of one arm.
Check students' drawings.
In their descriptions, students should refer to specific features of their drawings. The line dividing the 10:00 arm in half is a line of reflection, with the portion of the flake on each side being (nearly) a reflection of the other side. There’s a small imperfection in this description, with the large “ear” in the middle of the right side not quite having a mirror image where it should be. However, its almost-image on the other side can be created by reflecting the ear across the line of symmetry and then translating it slightly downward.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Ask students to write, say, or show sequences of rotations, reflections, and translations, using their hands. For example:
The entire flake can be created by rotating the arm through 60°, 120°, 180°, 240°, and 300°. For several of the new arms, the “ear” mentioned above appears in a slightly dilated form, or it appears several times as translations of one another. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Mark Cassino/Superstock
Reflection — both hands facing outward, thumbs nearly together. Translation — one hand facing forward and one backward in the same orientation. Rotation — one hand pointing left (thumb up) and one hand right (thumb down), fingers facing each other.
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CRITICAL THINKING Ask students whether a snowflake is a two-dimensional object. Have them consider the effects of the third dimension on lines of symmetry, and how the snowflake appears from a side view rather than a top view.
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Have students research the claim that “all snowflakes are different.” Depending upon students’ interests, the claim may lead them to investigate how and where snowflakes form, how they change as they fall through the atmosphere, why they have a hexagonal structure, and the effects that temperature and humidity have upon their structure.
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Sequences of Transformations
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LESSON
18.2
Name
Proving Figures are Congruent Using Rigid Motions
Date
18.2 Proving Figures are Congruent Using Rigid Motions Essential Question: How can you determine whether two figures are congruent? Resource Locker
Common Core Math Standards
Explore
The student is expected to: COMMON CORE
Class
Confirming Congruence
Two plane figures are congruent if and only if one can be obtained from the other by a sequence of rigid motions (that is, by a sequence of reflections, translations, and/or rotations).
G-CO.B.6
... given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. Also G-CO.A.5
A landscape architect uses a grid to design the landscape around a mall. Use tracing paper to confirm that the landscape elements are congruent.
Mathematical Practices
A
COMMON CORE
Trace planter ABCD. Describe a transformation you can use to move the tracing paper so that planter ABCD is mapped onto planter EFGH. What does this confirm about the planters?
MP.3 Logic
View the Engage section online. Discuss the photo and ask students to describe the pattern or patterns in the tile design. Then preview the Lesson Performance Task.
E H F
transformation that maps one to the other.
B
ENGAGE
G
Trace pools JKLM and NPQR. Fold the paper so that pool JKLM is mapped onto pool NPQR. Describe the transformation. What does this confirm about the pools?
J M K
You can map JKLM to NPQR with a reflection over the fold line. © Houghton Mifflin Harcourt Publishing Company
PREVIEW: LESSON PERFORMANCE TASK
C
down 4 units. The planters are congruent because there is a rigid
Have students work in pairs to label congruent and noncongruent figures.
Possible answer: If one figure can be obtained from the other by a sequence of rigid motions, then they are congruent.
B
You can map ABCD to EFGH with a translation left 4 units and
Language Objective
Essential Question: How can you determine whether two figures are congruent?
A D
P
L
The pools are congruent because there is a rigid transformation that maps one to the other.
C
N
Q
Determine whether the lawns are congruent. Is there a rigid transformation that maps △LMN to △DEF? What does this confirm about the lawns?
R
N
There is no sequence of rigid transformations that maps △DEF to △LMN. The lawns are not congruent.
M
D E
Reflect
1.
L F
How do the sizes of the pairs of figures help determine if they are congruent? If the figures are not the same size, there is no rigid motion that can map one of them onto
the other. The transformation would need to include a dilation, which is not a rigid motion.
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HARDCOVER PAGES 897908
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Watch for the hardcover student edition page numbers for this lesson.
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Determining if Figures are Congruent
Explain 1
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Students have the option of confirming congruence using the rigid motion activity either in the book or online.
The two figures appear to be the same/different.
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INTEGRATE TECHNOLOGY
You can map CDEF onto JKLM by reflecting CDEF over the y-axis. This reflection is a rigid motion that maps CDEF to JKLM, so the two figures are congruent. The coordinate notation for the reflection is (x, y) → (-x, y).
y X
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Confirming Congruence
The two figures appear to be the same size and shape, so look for a rigid transformation that will map one to the other.
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EXPLORE
Use the definition of congruence to decide whether the two figures are congruent. Explain your answer.
Example 1
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QUESTIONING STRATEGIES
B
You can map △ABC to △XYZ
x
by
A
This is/is not a rigid motion that maps △ABC to △XYZ, so the two figures are/are not congruent.
Do the figures appear to be congruent? Why or why not? yes, because they have the same size and shape
a counter-clockwise rotation of 90° around the origin .
x, y) → (-y, x) . The coordinate notation for the rotation is (
-8
Can either figure be considered to be the preimage? Why or why not? Yes, if ABCD is congruent to EFGH, then the reverse is true.
Your Turn
Use the definition of congruence to decide whether the two figures are congruent. Explain your answer. 2.
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EXPLAIN 1
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You can map ABCD to WXYZ with a reflection across the x-axis, so the figures are congruent. The coordinate notation for the reflection is (x, y) → (x, -y).
Module 18
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You can map △JKL to △XYZ with a reflection across the y-axis, followed by a horizontal translation, so the figures are congruent. The coordinate notation for the reflection is (x, y) → (-x, y) and for the translation is (x, y) → (x - 6, y).
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Determining if Figures are Congruent
© Houghton Mifflin Harcourt Publishing Company
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CONNECT VOCABULARY Define congruence. Ask students to give examples of congruent figures in the classroom. Students might mention floor tiles with the same size and shape, or desktops that are rectangles of the same size and shape. Tell students that “the same size and shape” is an informal way of deciding whether two figures may be congruent, but a formal definition of congruence is based on rigid motions.
Lesson 2
QUESTIONING STRATEGIES
PROFESSIONAL DEVELOPMENT IN1_MNLESE389762_U7M18L2 898
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Learning Progressions
In this lesson students learn that two figures are congruent if and only if there is a sequence of rigid motions that maps one figure to the other. That means if they can find the sequence of rigid motions that maps one figure to the other, then they can confirm that the preimage and image figures are congruent. It also means, if the figures are known to be congruent, that there is a sequence of rigid motions that maps one figure to the other. In upcoming lessons, students will use this transformations-based definition to develop congruence criteria for triangles.
How can a rigid motion be used to determine if two figures are congruent? Each rigid motion preserves size and shape, so if a sequence of rigid motions can be found to map one figure to the other, then the preimage and image figure are congruent.
Proving Figures are Congruent Using Rigid Motions
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EXPLAIN 2
The definition of congruence tells you that when two figures are known to be congruent, there must be some sequence of rigid motions that maps one to the other.
Finding a Sequence of Rigid Motions
Example 2
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Relate congruence to rigid motion by
Some students may think that if two figures are congruent, then there is one rigid motion that can map one figure to the other. Explain that it may take a sequence of rigid motions to map a figure to a congruent figure. Ask students to find examples of when this may be true.
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Map △ABC to △PQR with a rotation of 180° around the origin, followed by a horizontal translation.
Map ABCD to JKLM with a
Rotation: (x, y) → (-x, -y)
followed by a
reflection across the y-axis ,
Translation: (x, y) → (x + 1, y)
translation .
Reflection
: (x, y) →
(-x, y)
Translation : x, y → (x + 2, y - 10) ( )
Reflect
4.
© Houghton Mifflin Harcourt Publishing Company
AVOID COMMON ERRORS
8
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ABCD ≅ JKLM
y
8
QUESTIONING STRATEGIES
What would the notation (x, y) → (-x, y + 2) mean? The transformation reflects each x-coordinate across the x-axis and raises the figure by 2 units.
The figures shown are congruent. Find a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations you use.
△ABC ≅ △PQR
comparing the size and shape of the preimage and image. Point out that the sequence of rigid motions that maps one figure to another may not be unique. Encourage students to look for alternate sequences that work.
For each pair of figures, how do you know that a sequence of rigid motions that maps one figure to the other must exist? If the figures are known to be congruent—by the definition of congruent, there is a sequence of rigid motions that maps one to the other.
Finding a Sequence of Rigid Motions
Explain 2
How is the orientation of the figure affected by a sequence of transformations? If the transformations include a reflection, then the orientation will change. A translation
or rotation will preserve the original orientation. Your Turn
The figures shown are congruent. Find a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations you use. 5.
JKLM ≅ WXYZ M
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Reflect JKLM across the x-axis: (x, y) → (x, -y). Then translate the image: (x, y) → (x - 4, y - 2).
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ABCDE ≅ PQRST
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Reflect ABCDE across the y-axis: (x, y) → (-x, y). Then translate the image: (x, y) → (x, y - 10).
Lesson 2
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CONNECT VOCABULARY Help students understand how congruence is related to rigid motions by pointing out how a single rigid motion can produce a congruent figure. Therefore, a sequence of rigid motions must also produce a congruent figure. Point out that this is true both in a plane and on a coordinate plane.
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Lesson 18.2
Small Group Activity Give students the coordinates of a pair of congruent figures in the coordinate plane. Have each student describe a sequence of rigid motions that will map one figure to the other. Instruct them to switch papers and use another student’s sequence of rigid motions to confirm that the given figures are congruent. Have them use geometry software to do the rigid motions and check the results against their own sequences.
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Explain 3
Investigating Congruent Segments and Angles
EXPLAIN 3
Congruence can refer to parts of figures as well as whole figures. Two angles are congruent if and only if one can be obtained from the other by rigid motions (that is, by a sequence of reflections, translations, and/or rotations.) The same conditions are required for two segments to be congruent to each other. Example 3
Investigating Congruent Segments and Angles
Determine which angles or segments are congruent. Describe transformations that can be used to verify congruence.
QUESTIONING STRATEGIES
F C
A
How does the congruence of angles and segments relate to the congruence of two figures? Why? Since rigid motions preserve angle measure and distance, verifying that corresponding angles and corresponding segments have the same measure determines whether two figures are congruent.
E A
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_ AB and CD are congruent. A sequence of transformations is a reflection and a translation.
∠A and ∠C are congruent. The transformation is a translation. There is no transformation that maps ∠B to either of the other angles.
There is no transformation that maps to either of the other segments.
―
EF
Your Turn
7.
Determine which segments and which angles are congruent. Describe transformations that can be used to show the congruence.
―
AVOID COMMON ERRORS C
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∠B and ∠C are congruent. EF and GH are congruent. In both cases, a sequence of transformations is a reflection and a translation.
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Students may believe that two angles cannot be congruent if the rays forming the angles have different lengths. Remind students that rays continue forever in one direction, so the length representing a ray in a diagram is arbitrary. Draw two congruent angles, one with longer rays. Discuss why the angles are congruent even though one appears to be larger.
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8.
Can you say two angles are congruent if they have the same measure but the segments that identify the rays that form the angle are different lengths? Yes, the definition of congruence for angles requires only that the angle between the rays
be the same. The lengths of the segments does not matter. 9.
Discussion Can figures have congruent angles but not be congruent figures? Yes, two figures can have congruent angles but not be congruent figures. They could appear to be different sized versions of the same figure.
© Houghton Mifflin Harcourt Publishing Company
Elaborate
ELABORATE QUESTIONING STRATEGIES Can you say two angles are congruent if they have the same measure but the segments that identify the rays that form the angle are different lengths? Explain. Yes. The angle measures determine if the two angles are congruent, not the rays or parts of the rays that make up their sides.
10. Essential Question Check-In Can you use transformations to prove that two figures are not congruent? Maybe. If a dilation with scale factor ≠ 1 maps one figure onto the other, then the figures
cannot not be mapped using only rigid motions, so they cannot be congruent. Module 18
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Visual Cues Draw two congruent triangles and label the vertices. Highlight one side of one triangle blue. Have students name the corresponding side of the other triangle and highlight that side blue as well. Repeat with the other two pairs of corresponding sides using green and purple. Then shade one of the angles red. Have students name the corresponding angle of the other triangle and shade that red as well. Repeat with the other two pairs of corresponding angles using yellow and orange.
Can you say two segments are congruent if their orientation is different? Explain. Yes. The orientation of the segments does not affect their lengths, and therefore does not affect their congruence.
SUMMARIZE THE LESSON How are congruent figures related to transformations? Two figures are congruent if one can be mapped to the other by a rigid transformation (rotation, reflection, or translation) or by a sequence of rigid transformations.
Proving Figures are Congruent Using Rigid Motions
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Evaluate: Homework and Practice
EVALUATE
• Online Homework • Hints and Help • Extra Practice
Use the definition of congruence to decide whether the two figures are congruent. Explain your answer. Give coordinate notation for the transformations you use. 1. E
ASSIGNMENT GUIDE Concepts and Skills
-8
Exercises 1–5, 14–15, 17
Example 2 Finding a Sequence of Rigid Motions
Exercises 6–9, 16, 18–20, 24, 26–31
Example 3 Investigating Congruent Segments and Angles
Exercises 10–13, 23, 25
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There is no sequence of rigid transformations that will map one figure onto the other, so they are not congruent.
There is no sequence of rigid transformations that will map one figure onto the other, so they are not congruent.
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You can map DEFG to WXYZ with a reflection across the x-axis, followed by a horizontal translation. So, the two figures are congruent. reflection: (x, y) →( x, -y); translation: (x, y) → (x - 10, y).
You can map ABCDE to PQRST with a translation. So, the figures are congruent. translation: (x, y) → (x - 2, y - 7). 5.
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congruent by using geometry software to do a sequence of rigid motions. Remind students to use the measuring features to show that angle measures and segment lengths are preserved.
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You can map △CDE onto △JKL by reflecting △CDE over the x-axis, followed by a horizontal translation. So, the two figures are congruent. reflection: (x, y) → (x, -y); translation: (x, y) → (x + 8, y). 3.
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INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can verify that two figures are
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Example 1 Determining if Figures are Congruent
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The figures shown are congruent. Find a sequence of rigid motions that maps one figure to the other. Give coordinate notation for the transformations you use. 6.
RSTU ≅ WXYZ U
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△ABC ≅ △DEF
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DEFGH ≅ PQRST
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Map △ABC to △DEF with a rotation of 180° around the origin, followed by a translation. rotation: (x, y) → (-x, -y); translation: (x, y) → (x + 2, y + 6)
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investigate which corresponding segments for two figures are congruent and which pairs of corresponding angles are congruent. Have them fold the tracing paper to see if the figures are coincident and, if they are, then write a sequence of rigid motions that can map one figure to the other. Have them also use the tracing paper to draw the figure and its congruent image on graph paper and then give the algebraic rules that map one figure to the other.
A x 8
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Map RSTU to WXYZ with a reflection across the y-axis, followed by a translation. reflection: (x, y) → (-x, y); translation: (x, y) → (x + 1, y - 4)
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INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Suggest that students use tracing paper to
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Determine which of the angles are congruent. Which transformations can be used to verify the congruence? 10.
11. A
A B
B
© Houghton Mifflin Harcourt Publishing Company
Map △CDE to △WXY with a rotation of 180° around the origin, followed by a horizontal translation. rotation: (x, y) → (-x, -y); translation: (x, y) → (x - 2, y)
Map DEFGH to PQRST with a reflection across the y-axis, followed by a vertical translation. reflection: (x, y) → (-x, y); translation: (x, y) → (x, y - 8)
C
C
∠A, ∠B and ∠C are all congruent. The sequence of transformations is a reflection and a translation.
None of the angles are congruent. There is no transformation that maps one of the angles to another. Module 18
Exercise
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Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
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Proving Figures are Congruent Using Rigid Motions
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Determine which of the segments are congruent. Which transformations can be used to verify the congruence?
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 When examining congruent figures on graph
A
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13.
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C B
paper, students can see how each vertex is mapped to its corresponding vertex by circling corresponding vertices in the same color, using a different color for each pair of corresponding vertices. Students can also highlight pairs of corresponding sides in the same color, using a different color for each pair.
F
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AB and CD are congruent; reflection, then translation. There is no transformation that ― maps EF to either of the other segments.
None of the segments are congruent. There is no rigid transformation that maps one of them to another.
Use the definition of congruence to decide whether the two figures are congruent. Explain your answer. Give coordinate notation for the transformations you use. 14.
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Yes. Map EFGH to RSTU with a counter-clockwise rotation of 90° around the origin, followed by a vertical translation. rotation: (x, y) → (-y, x) translation: (x, y) → (x, y + 4).
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Yes. Map BCDE to JKLM with a reflection across the x-axis, followed by a horizontal translation. reflection: (x, y) → (x, -y) translation: (x, y) → (x + 4, y).
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Yes. Map △JKL to △WXY with a clockwise rotation of 90° around the origin, followed by a translation. rotation: (x, y) → (y, -x) translation: (x, y) → (x + 1, y + 6). y R 8
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No, the figures are not congruent. There are no transformations to map △KLM to △WXY.
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The figures shown are congruent. Find a sequence of transformations for the indicated mapping. Give coordinate notation for the transformations you use.
AVOID COMMON ERRORS
18. Map DEFGH to PQRST.
Students may make an error when using computations to determine if a transformed figure is congruent or not congruent. Emphasize that a resulting figure with the sides crossing each other is an indication of an error, not necessarily a noncongruent figure.
y 8 F S
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Map JKLM to WXYZ with a reflection across the y-axis, followed by a vertical translation. reflection: (x, y) → (-x, y); translation: (x, y) → (x, y - 6).
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20. Map ABCDEF to PQRSTU.
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Map DEFGH to PQRST with a rotation of 180° around the origin. The coordinate notation for the rotation is (x, y) → (-x, -y).
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19. Map JKLM to WXYZ.
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Map △DEF to △KLM with a rotation of 180° around the origin, followed by a horizontal translation. Rotation: (x, y) → (-x, -y); translation: (x, y) → (x, - 4, y).
22. Determine whether each pair of angles is congruent or not congruent. Select the correct answer for each lettered part.
a. ∠A and ∠B b. ∠A and ∠C
c. ∠B and ∠C d. ∠B and ∠D
e. ∠C and ∠D
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Congruent Congruent
Not congruent Not congruent
Congruent Congruent
Not congruent Not congruent
Congruent
Not congruent
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© Houghton Mifflin Harcourt Publishing Company
Map ABCDEF to PQRSTU with a combined translation. The coordinate notation for the translation is (x, y) → (x - 6, y - 10).
Lesson 2
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Proving Figures are Congruent Using Rigid Motions
904
23. If ABCD and WXYZ are congruent, then ABCD can be mapped to WXYZ using a rotation and a translation. Determine whether the statement is true or false. Then explain your reasoning.
COLLABORATIVE LEARNING Give each student a sheet of graph paper. On the top half, have students draw 4 ABC. Then ask them to perform a sequence of two or three rigid motions to draw 4 A'B'C'. They may use each transformation only once. On the bottom half, have them write the sequence of rigid motions using precise mathematical language or symbols, then cut the paper in half. Collect the half sheets, making one pile of drawings and one pile of descriptions. Randomly pass out the papers so that each student receives one from each pile. Students should try to match each drawing with its corresponding rigid motions.
Y D W
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False. The figures do not have the same orientation, so the sequence of transformations must include a reflection.
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24. Which segments are congruent? Which are not congruent? Explain.
None are congruent. No rigid motions map one segment onto another.
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25. Which angles are congruent? Which are not congruent? Explain.
None are congruent. No rigid motions map one angle onto another.
D B A C
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26. The figures shown are congruent. Find a sequence of transformations that will map CDEFG to QRSTU. Give coordinate notation for the transformations you use. 8
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27. The figures shown are congruent. Find a sequence of transformations that will map △LMN to △XYZ. Give coordinate notation for the transformations you use.
Rotate CDEFG 90° clockwise about the origin: (x, y) → (-y, x). Then reflect across the x-axis:
8 4
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(x, y) → (x, -y).
Y
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(x, y) → (x - 4, y - 9 ).
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Then translate:
Rotate △LMN 90° counter-clockwise about the origin:
y
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(x, y) → (-y, x). Reflect across the y-axis: (x, y) → (-x, y). Translate: (x, y) → (x - 3, y - 13).
28. Which sequence of transformations does not map a figure onto a congruent figure? Explain. A. Rotation of 180° around the origin, reflection across the x-axis, horizontal translation (x, y) → (x + 4, y)
B. Reflection across the y-axis, combined translation (x, y) → (x -5, y + 2) C. Rotation of 180° around the origin, reflection across the y-axis,dilation (x, y) → (2x, 2y) A dilation is not a rigid transformation.
D. Counterclockwise rotation of 90° around the origin, reflection across the y-axis, combined translation (x, y) → (x -11, y - 12)
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29. The figures shown are congruent. Find a sequence of transformations that will map DEFGH to VWXYZ. Give coordinate notation for the transformations you use.
Map DEFGH to VWXYZ with a clockwise rotation of 90° around the origin, followed by a reflection across the y-axis, followed by a combined translation. rotation: (x, y) → (y, -x); reflection: (x, y) → (-x, y); translation: (x, y) → (x + 2, y - 9).
8 H G -8
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INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Be sure that students’ answers are as detailed
4
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and precise as possible. When stating that two figures in the coordinate plane are congruent, students should describe the specific rigid motion(s) that map one figure to the other. For instance, “a reflection across the x-axis” or “the reflection (x, y) → (x, −y)” are detailed descriptions of a rigid motion in the coordinate plane, while “reflection” is not.
X
30. How can you prove that two arrows in the recycling symbol are congruent to each other?
The arrows can each be mapped to each other by a rotation, which is a rigid transformation.
31. The city of St. Louis was settled by the French in the mid 1700s and joined the United States in 1803 as part of the Louisiana Purchase. The city flag reflects its French history by featuring the fleur-de-lis. How can you prove that the left and right petals are congruent to each other?
The petals can be mapped onto each other by a reflection, which is a rigid transformation.
Only the first student is correct. The two figures have the same orientation, so a sequence of transformations including a single reflection will change the orientation of the result.
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32. Draw Conclusions Two students are trying to show that the two figures are congruent. The first student decides to map CDEFG to PQRST using a rotation of 180° around the origin, followed by the translation (x, y) → (x, y + 6). The second student believes the correct transformations are a reflection across the y-axis, followed by the vertical translation (x, y) → ( x, y - 2 ). Are both students correct, is only one student correct, or is neither student correct?
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Proving Figures are Congruent Using Rigid Motions
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33. Justify Reasoning Two students are trying to show that the two figures are congruent. The first student decides to map DEFG to RSTU using a rotation of 180° around the origin, followed by the vertical translation (x, y) → (x, y + 4). The second student uses a reflection across the x-axis, followed by the vertical translation (x, y) → (x, y + 4), followed by a reflection across the y-axis. Are both students correct, is only one student correct, or is neither student correct?
PEERTOPEER DISCUSSION Have students discuss with a partner how to predict the sequence of rigid motions that may map a figure to a congruent figure. Then have them predict and test whether reversing the order of the sequence of rigid motions will produce the preimage figure.
Both students are correct. Either of the sequences of transformation will map DEFG to RSTU. Recall that a rotation of 180° around the origin is the same as a reflection across both axes.
CONNECT VOCABULARY
-8
-4
4 U 0 -4
G E
Fx 8
D
H.O.T. Focus on Higher Order Thinking
34. Look for a Pattern Assume the pattern of congruent squares shown in the figure continues forever. y 4 Write rules for rigid motions that map square 0 0 onto square 1, square 0 onto square 2, and square 0 onto square 3. 1 x (x, y) → (x + 2, y - 2) -4 -2 0 2 4 (x, y) → (x + 4, y - 4) -2 (x, y) → (x + 6, y - 6) 3 Write a rule for a rigid motion that maps square 0 -4 onto square n.
JOURNAL Have students write a journal entry in which they summarize what they know so far about congruence. Prompt students to include examples and non-examples of congruent figures and the methods of obtaining them and determining them. © Houghton Mifflin Harcourt Publishing Company
(x, y) → (x + 2n, y - 2n) 35. Analyze Relationships Suppose you know that △ABC is congruent to △DEF and that △DEF is congruent to △GHJ. Can you conclude that △ABC is congruent to △GHJ? Explain.
Yes; by the definition of congruence, there is a sequence of rigid motions that maps △ABC onto △DEF and another that maps △DEF onto △GHJ. The first sequence followed by the second sequence maps △ABC onto △GHJ, so the triangles are congruent.
Ideas Ella plotted the points A(0, 0), B(4, 0), and 36. Communicate Mathematical _ _ C(0, 4). Then she drew AB and AC. Give two different arguments to explain why the segments are congruent. Both segments are 4 units long. Because the segments are the same ¯. ¯ onto AC length, they are congruent. A rotation of 90°△ maps AB Because there is a rigid motion that maps one segment onto the other, the segments are congruent.
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Have students relate the word congruent to the terms equal and equivalent. If a figure is congruent to another it is equal in shape and size. Show students similar figures that are equal in shape but not size, and discuss why they are not congruent.
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Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Ask students to show how each of the nine
The illustration shows how nine congruent shapes can be fitted together to form a larger shape. Each of the shapes can be formed from Shape #1 through a combination of translations, reflections, and/or rotations.
pieces in the Lesson Performance Task can be divided into two congruent shapes so that the entire shape can be constructed from 18 congruent shapes. The shapes are isosceles trapezoids that form an L when placed together.
2 1
3 4
5
INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 An object viewed through certain types of
7 6
8
9
Describe how each of Shapes 2–9 can be formed from Shape #1 through a combination of translations, reflections, and/or rotations. Then design a figure like this one, using a least eight congruent shapes. Number the shapes. Then describe how each of them can be formed from Shape #1 through a combination of translations, reflections, and/or rotations.
lenses will appear to be flipped upside-down. Ask why the letters in the word STAR are flipped when seen through such a lens but the word CODE is not. The letters C, O, D, and E are symmetric with respect to a line drawn horizontally through their centers, while the letters S, T, A, and R are not. The result is that CODE is indeed “flipped” by the lens, and the image through the lens appears exactly as it did before. The same is not true of the letters of STAR.
Shape #2: Rotate Shape #1 180°. Shape #3: Reflect Shape #1 vertically. Shape #4: Translate Shape #1 down and right. Shape #5: Rotate Shape #1 180° and then translate it down and right. © Houghton Mifflin Harcourt Publishing Company
Shape #6: Translate Shape #1 down. Shape #7: Translate Shape #1 down and right. Shape #8: Translate Shape #1 down and then reflect it horizontally. Shape #9: Translate Shape #1 down and right.
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The broken lines on the figure show how it can be divided into three congruent isosceles right triangles. Have students copy the figure and determine how it can be divided into eight congruent trapezoids.
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Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Proving Figures are Congruent Using Rigid Motions
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LESSON
18.3
Name
Corresponding Parts of Congruent Figures Are Congruent
Class
Date
18.3 Corresponding Parts of Congruent Figures Are Congruent Essential Question: What can you conclude about two figures that are congruent?
Resource Locker
Common Core Math Standards The student is expected to: COMMON CORE
Explore
G-CO.B.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
You will investigate some conclusions you can make when you know that two figures are congruent.
A
Mathematical Practices COMMON CORE
Exploring Congruence of Parts of Transformed Figures
Fold a sheet of paper in half. Use a straightedge to draw a triangle on the folded sheet. Then cut out the triangle, cutting through both layers of paper to produce two congruent triangles. Label them △ ABC and △ DEF, as shown.
MP.2 Reasoning A
Language Objective Have students fill in sentence stems to explain why figures are congruent or noncongruent.
B D
C
E
F
ENGAGE
The corresponding parts are congruent, and relationships within the figures, such as relative distances between vertices, are equal.
© Houghton Mifflin Harcourt Publishing Company
Essential Question: What can you conclude about two figures that are congruent?
B
A translation (perhaps followed by a rotation) maps △ABC to △DEF.
C
The same sequence of rigid motions that maps △ ABC to △ DEF maps parts of △ ABC to parts of △DEF. Complete the following. _ _ _ AB → ¯ BC → ¯ AC → ¯ EF DF DE A
D
PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo and ask students to identify congruent shapes in the design. Then preview the Lesson Performance Task.
Place the triangles next to each other on a desktop. Since the triangles are congruent, there must be a sequence of rigid motions that maps △ ABC to △ DEF. Describe the sequence of rigid motions.
→
D
B
→
→
C
E
F
What does Step C tell you about the corresponding parts of the two triangles? Why?
The corresponding parts are congruent because there is a sequence of rigid motions that maps each side or angle of △ ABC to the corresponding side or angle of △ DEF.
Module 18
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congruent? s that are es are two figure that two triangl de about s to show are you conclu What can of rigid motion pairs of angles Question: ence in terms corresponding Essential ion of congru pairs of sides and the definit COMMON G-CO.B.7 Use if corresponding sformed CORE if and only s of Tran congruent congruent.
Exploring Figures
HARDCOVER PAGES 909920
e of Part Congruenc
Watch for the hardcover student edition page numbers for this lesson.
ent. are congru two figures know that when you can make sheet. the folded uent triangle on two congr to draw a to produce a straightedge layers of paper half. Use of paper in through both Fold a sheet le, cutting shown. out the triang BC and △ DEF, as A Then cut △ Label them triangles.
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Reflect
EXPLORE
If you know that △ ABC ≅ △ DEF, what six congruence statements about segments and angles can you write? Why? ¯ AB ≅ ¯ DE, ¯ BC ≅ ¯ EF, ¯ AC ≅ ¯ DF, ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F. The rigid motions that map
1.
Exploring Congruence of Parts of Transformed Figures
△ABC to △DEF also map the sides and angles of △ABC to the corresponding sides and angles of △DEF, which establishes congruence. Do your findings in this Explore apply to figures J K other than triangles? For instance, if you know that P Q quadrilaterals JKLM and PQRS are congruent, can M L you make any conclusions about corresponding S parts? Why or why not? Yes; since quadrilateral JKLM is congruent to quadrilateral PQRS, there is a sequence of
2.
QUESTIONING STRATEGIES When you are given two congruent triangles, how many pairs of corresponding parts—angles and sides—are there? 6; 3 angles and 3 sides
R
rigid motions that maps JKLM to PQRS. This same sequence of rigid motions maps sides and angles of JKLM to the corresponding sides and angles of PQRS.
Explain 1
Corresponding Parts of Congruent Figures Are Congruent
EXPLAIN 1
The following true statement summarizes what you discovered in the Explore.
Corresponding Parts of Congruent Figures Are Congruent
Corresponding Parts of Congruent Figures Are Congruent If two figures are congruent, then corresponding sides are congruent and corresponding angles are congruent. Example 1
DE
D A
3.5 cm
2.6 cm
F
B
DE = AB, and AB = 2.6 cm, so DE = 2.6 cm.
3.7 cm
42°
73° 65°
E
C
m∠B Step 1 Find the angle that corresponds to ∠B. Since △ABC ≅ △DEF, ∠B ≅ ∠ E .
© Houghton Mifflin Harcourt Publishing Company
_ Step 1 Find the side that corresponds to DE . _ _ Since △ABC ≅ △DEF, AB ≅ DE .
Step 2 Find the unknown length.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have a student read the Corresponding Parts
△ ABC ≅ △ DEF. Find the given side length or angle measure.
of Congruent Figures Theorem. Discuss the meaning of the theorem for general figures and then in terms of two triangles. Emphasize that the theorem is biconditional, an if-and-only-if statement that is true when read as an if-then statement in either direction.
Step 2 Find the unknown angle measure. m∠B = m∠ E , and m∠ E = 65 °, so m∠B = 65 °.
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Math Background
In this lesson, students learn that if two figures (including triangles) are congruent, then corresponding pairs of sides and corresponding pairs of angles of the figures are congruent. This follows readily from the rigid-motion definition of congruence and from the Corresponding Parts of Congruent Figures Are Congruent Theorem. This theorem is biconditional, a statement that is true in either direction. That is, if corresponding pairs of sides and corresponding pairs of angles in two figures are congruent, then the figures are congruent.
Corresponding Parts of Congruent Figures are Congruent
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Reflect
QUESTIONING STRATEGIES
Discussion _ _The triangles shown in the figure are congruent. Can you conclude that JK ≅ QR? Explain. K No; the segments appear to be congruent, but the
3.
How do you determine which sides of two congruent figures correspond? Use the order of letters in the congruence statement. The first letters correspond, the last letters correspond, and the other letters correspond in the same order.
correspondence between the triangles is not given,
Q
J
VISUAL CUES
parts.
P
L
Your Turn
△STU ≅ △VWX. Find the given side length or angle measure.
Have each student make a poster illustrating the concept of congruent figures. The illustrations should be labeled to show which pairs of corresponding parts are congruent. Have them show both examples and non-examples of congruent figures in the poster.
X
S
32 ft
124°
EXPLAIN 2
5. m∠S Since △STU ≅ △VWX, ∠S ≅ ∠V.
38°
m∠S = m∠V = 38°.
Applying the Properties of Congruence
Rigid motions preserve length and angle. This means that congruent segments have the _ _ same length, so UV ≅ XY implies UV = XY and vice versa. In the same way, congruent angles have the same measure, so ∠J ≅ ∠K implies m∠J = m∠K and vice versa.
Properties of Congruence
_ _ AB ≅ AB _ _ _ _ If AB ≅ CD , then CD ≅ AD . _ _ _ _ _ _ If AB ≅ CD and CD ≅ EF , then AB ≅ EF .
© Houghton Mifflin Harcourt Publishing Company
Symmetric Property of Congruence Transitive Property of Congruence Example 2
Lesson 18.3
△ABC ≅ △DEF. Find the given side length or angle measure.
AB
B
_ _ Since △ABC ≅ △DEF, AB ≅ DE . Therefore, AB = DE. Write an equation.
QUESTIONING STRATEGIES
911
SU = VX = 43 ft.
43 ft
T
Reflexive Property of Congruence
How could you use transformations to decide whether two figures are congruent? You could use transformations to create all pairs of corresponding parts congruent. Then the theorem applies because if corresponding parts of congruent figures are congruent, then the figures are congruent.
SU ≅ ¯ VX. Since △STU ≅ △VWX, ¯
V
Explain 2
Applying the Properties of Congruence
that relate the parts of the figures and mark the figures to show them. Once they have clearly represented the corresponding parts, they can more easily answer the questions.
4. SU
18°
W
16 ft
U
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 Suggest that students list all the congruencies
so you cannot assume ¯ JK and ¯ QR are corresponding
R
(3x + 8) in. A
3x + 8 = 5x
Subtract 3x from each side.
8 = 2x
Divide each side by 2.
4=x
(5y + 11)°
(6y + 2)° 25 in. C
F
D (5x) in. 83° E
So, AB = 3x + 8 = 3(4) + 8 = 12 + 8 = 20 in. Module 18
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Small Group Activity
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Have each student draw a pair of congruent figures on paper. Instruct them to switch papers and to write a congruence statement for the pair of figures. Then have them switch papers several more times within groups, write new congruence statements that fit the pair of figures, and list the congruent pairs of corresponding parts of the figures.
m∠D
AVOID COMMON ERRORS
Since △ABC ≅ △DEF, ∠ A ≅ ∠D. Therefore, m∠ A = m∠ D.
Students may correctly solve for a variable but then incorrectly give the value of the variable as a side length or angle measure. Remind them to examine the diagram carefully; sometimes a side length or angle measure is described by an expression containing a variable, not by the variable alone.
5y + 11 = 6y + 2
Write an equation. Subtract 5y from each side.
11 = y + 2
Subtract 2 from each side.
9 = y
(
)
So, m∠D = (6y + 2)° = 6 ⋅ 9 + 2 ° = 56 ° . Your Turn
Quadrilateral GHJK ≅ quadrilateral LMNP. Find the given side length or angle measure. G
(4x + 3) cm H (9y + 17)°
P
J
(11y - 1)°
N
LM
¯ ≅ LM ¯. Since GHJK ≅ LMNP, GH Therefore, GH = LM.
7.
4x + 3 = 6x - 13 → 8 = x
INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Encourage students to use geometry software
m∠H
9y + 17 = 11y - 1 → 9 = y
m∠H = (9y + 17)° = (9 ⋅ 9 + 17)° = 98°
Using Congruent Corresponding Parts in a Proof
Write each proof.
A
Given: △ABD ≅ △ACD
_ Prove: D is the midpoint of BC.
B
Statements 1. △ABD ≅ △ACD _ _ 2. BD ≅ CD _ 3. D is the midpoint of BC .
Module 18
D
C
Reasons 1. Given 2. Corresponding parts of congruent figures are congruent.
to reflect the triangle with the given conditions and then to verify that corresponding congruent parts have equal measure.
© Houghton Mifflin Harcourt Publishing Company
Example 3
M
Since quadrilateral GHJK ≅ quadrilateral LMNP, ∠H ≅ ∠M. Therefore, m∠H = m∠M.
LM = 6x - 13 = 6(8) - 13 = 35 cm
Explain 3
Using Congruent Corresponding Parts in a Proof
(6x − 13) cm
K
6.
EXPLAIN 3
18 cm L (10y)°
CONNECT VOCABULARY In this lesson, students learn the Corresponding Parts of Congruent Figures Are Congruent Theorem. Although acronyms (such as CPCTC) may be helpful to some students when referring to theorems, such devices may be a bit more difficult for English Learners at the Emerging level. Consider making a poster or having students create or copy a list of theorems, along with their meanings, for them to refer to in this module. Students may want to come up with a mnemonic for the CPCTC itself, such as Cooks Pick Carrots Too Carefully.
3. Definition of midpoint.
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Technology Have students use geometry software to create designs using congruent triangles. They should arrange multiple congruent triangles using different colors, positions, and orientations. Ask them to make three separate designs: one using congruent equilateral triangles, one using congruent isosceles triangles, and one using congruent scalene triangles.
Corresponding Parts of Congruent Figures are Congruent
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Given: Quadrilateral JKLM ≅ quadrilateral NPQR; ∠J ≅ ∠K
B
QUESTIONING STRATEGIES
Prove: ∠J ≅ ∠P
Why do pairs of corresponding congruent parts have equal measure? Since rigid motions preserve angle measure and length, and since there is a sequence of rigid motions that maps a figure to a congruent figure, pairs of corresponding parts must have equal measure.
Statements
8.
Suppose you know that 4CBA ≅ 4EFG. What are six congruency statements? ∠C ≅ ¯ ≅ EF ¯, CA ¯ ≅ EG ¯, BA ¯ ≅ FG ¯ ∠E, ∠B ≅ ∠F, ∠A ≅ ∠G, CB
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Lesson 18.3
Reasons
1. Quadrilateral JKLM ≅ quadrilateral NPQR
1. Given
2. ∠J ≅ ∠K
2. Given
3. ∠K ≅ ∠P
3. Corresponding parts of congruent
4. ∠J ≅ ∠P
4. Transitive Property of Congruence
figures are congruent.
V
Given: △SVT ≅ △SWT _ Prove: ST bisects ∠VSW.
S
Statements
T
Reasons
1. △ SVT ≅ △ SWT
1. Given
2. ∠VST ≅ ∠WST
2. Corresponding parts of congruent figures are congruent.
3. ¯ ST bisects ∠VSW. 9.
3. Definition of angle bisector.
Given: Quadrilateral _ _ ABCD ≅ quadrilateral EFGH; AD ≅ CD _ _ Prove: AD ≅ GH
B
A
D
Statements
C
F
E
H
G
Reasons
1. Quadrilateral ABCD ≅ quadrilateral EFGH
2. ¯ AD ≅ ¯ CD
1. Given 2. Given
3. ¯ CD ≅ ¯ GH
3. Corresponding parts of congruent
4. ¯ AD ≅ ¯ GH
4. Transitive Property of Congruence
figures are congruent.
Module 18
SUMMARIZE THE LESSON
P
W
© Houghton Mifflin Harcourt Publishing Company
Can you say that a pair of corresponding sides of two congruent figures has equal measure? Yes. If the figures are congruent, then each pair of corresponding sides is congruent and therefore has equal measure.
N
Write each proof.
can see how each vertex is mapped to its corresponding vertex by designating corresponding vertices in the same color and using a different color for each pair of corresponding vertices. Students can also highlight pairs of corresponding sides in the same color, using a different color for each pair.
Can you say two figures are congruent if their corresponding angles have the same measure? Explain. No. You must also determine that the corresponding sides have the same measure.
Q
Your Turn
INTEGRATE MATHEMATICAL PRACTICES Focus on Modeling MP.4 When examining congruent figures, students
QUESTIONING STRATEGIES
L M
ELABORATE
R
K
J
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Connect Vocabulary Have students work in pairs. Provide each student with a protractor and ruler, and ask them to explain why two figures are congruent or noncongruent. Provide students with sentence stems to help them describe the attributes of the figures. For example: “The two (triangles/quadrilaterals/figures) are or are not congruent because their corresponding angles have/don’t have equal measures. Angles ___ and ____ are corresponding, and measure _____ degrees. Corresponding sides have equal/not equal lengths.” Students work together to complete the sentences.
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Elaborate
EVALUATE
10. A student claims that any two congruent triangles must have the same perimeter. Do you agree? Explain. Yes; since the corresponding sides of congruent triangles are congruent, the sum of the
lengths of the sides (perimeter) must be the same for both triangles. 11. If △PQR is a right triangle and △ PQR ≅ △ XYZ, does △ XYZ have to be a right
triangle? Why or why not? Yes; since △ PQR is a right triangle, one of its angles is a right angle. Since corresponding
parts of congruent figures are congruent, one of the angles of △ XYZ must also be a right
ASSIGNMENT GUIDE
angle, which means △ XYZ is a right triangle. 12. Essential Question Check-In Suppose you know that pentagon ABCDE is congruent to pentagon FGHJK. How many additional congruence statements can you write using corresponding parts of the pentagons? Explain. There are five statements using the congruent corresponding sides and five statements
using the congruent corresponding angles.
Evaluate: Homework and Practice 1.
Danielle finds that she can use a translation and a reflection to make quadrilateral ABCD fit perfectly on top of quadrilateral WXYZ. What congruence statements can Danielle write using the sides and angles of the quadrilaterals? Why? A B
• Online Homework • Hints and Help • Extra Practice
Y
W
The same sequence of rigid motions that maps ABCD to WXYZ also maps sides and angles of ABCD to corresponding sides and angles of WXYZ. Therefore, those sides and angles are _ congruent: ¯ AB ≅ ¯ WX, ¯ BC ≅ ¯ XY, CD ≅ ¯ YZ, ¯ AD ≅ ¯ WZ, ∠A ≅ ∠W, ∠B ≅ ∠X, ∠C ≅ ∠Y, ∠D ≅ ∠Z. △DEF ≅ △GHJ. Find the given side length or angle measure. G
D 42 ft
19 ft
2.
JH
112°
¯ ≅ JH ¯. Since △ DEF ≅ △ GHJ, FE FE = JH = 31 ft, so JH = 31 ft.
Module 18
Exercise
IN1_MNLESE389762_U7M18L3 914
F
31 ft
3.
Exercises 1
Example 1 Corresponding Parts of Congruent Figures are Congruent
Exercises 2–5, 10–13
Example 2 Applying the Properties of Congruence
Exercises 6–9
Example 3 Using Congruent Corresponding Parts in a Proof
Exercises 14–16
J
25°
43°
quadrilaterals, both with side lengths of 1 foot on each side, are congruent. Students should recognize that the description is that of a rhombus. Demonstrate that a box with an open top and bottom lying on its side is not rigid, and although the side lengths stay the same when one side is pushed, the angles change. Thus it is possible for the two figures described to have different angle measures and not be congruent.
H
m∠D Since △ DEF ≅ △ GHJ, ∠D ≅ ∠G. m∠D = m∠G = 43° Lesson 3
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Depth of Knowledge (D.O.K.)
Explore Exploring Congruence of Parts of Transformed Figures
INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Have students consider whether two
© Houghton Mifflin Harcourt Publishing Company
X
E
Practice
Z
C
D
Concepts and Skills
COMMON CORE
Mathematical Practices
1
1 Recall of Information
MP.6 Precision
2–5
1 Recall of Information
MP.2 Reasoning
6–9
1 Recall of Information
MP.4 Modeling
14–16
2 Skills/Concepts
MP.3 Logic
10–13, 17–18
2 Skills/Concepts
MP.2 Reasoning
19–22
2 Skills/Concepts
MP.4 Modeling
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Corresponding Parts of Congruent Figures are Congruent
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KLMN ≅ PQRS. Find the given side length or angle measure.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Have students compare their congruence
K 2.1 cm N
statements for a given diagram, and ask them to write other correct congruence statements for the same diagram. Then have them write a congruence statement that is not correct for the diagram and explain why it is not correct.
S
L
R
79° M
75°
P
2.9 cm Q
4.
m∠R ∠M ≅ ∠R. m∠M = m∠R = 79°.
PS ¯ KN ≅ ¯ PS. KN = PS = 2.1 cm
5.
△ ABC ≅ △ TUV. Find the given side length or angle measure. A (3y + 2)°
C
(6x + 2) cm
(5x + 7) cm
V
(6x - 1) cm
(4y)°
B
T
¯. So, BC = UV. BC ¯ BC ≅ UV 6x + 2 = 5x + 7 → x = 5
6.
U (4y - 18)°
m∠U ∠B ≅ ∠U. So, m∠B ≅ m∠U. 3y + 2 = 4y - 18 → 20 = y
7.
So, m∠U = (4y - 18)° = (4 ⋅ 20 - 18)° = 62°.
So, BC = 6x + 2 = 6(5) + 2 = 30 + 2 = 32 cm.
DEFG ≅ KLMN. Find the given side length or angle measure. N
D (20x + 12)°
68°
E F
© Houghton Mifflin Harcourt Publishing Company
G
(2y + 3)in.
K
(4y - 29) in. M
(25x - 8)°
FG ¯ FG ≅ ¯ MN. So, FG = MN. 2y + 3 = 4y - 29 → 16 = y
8.
9.
_ ¯ 10. GH ≅ ST
11. ∠J ≅ ∠U
△ GHJ ≅ △ STU by the Transitive Prop. of
△ GHJ ≅ △ STU by the Transitive Prop. of
Cong., and corr. parts of ≅ fig. ≅.
Cong., and corr. parts of ≅ fig. ≅.
13. m∠G = m∠G
12. GJ = SU
△ GHJ ≅ △ STU by the Transitive Prop.
of Cong., and corr. parts of ≅ fig. ≅.
of Cong., and corr. parts of ≅ fig. ≅.
Congruent segments have the same length,
Congruent angles have the same
so GJ = SU.
measure, so m∠G = m∠S.
Module 18
Exercise
IN1_MNLESE389762_U7M18L3 915
Lesson 3
915
Depth of Knowledge (D.O.K.)
COMMON CORE
Mathematical Practices
2 Skills/Concepts
MP.2 Reasoning
24–25
3 Strategic Thinking
MP.3 Logic
26
3 Strategic Thinking
MP.6 Precision
27
3 Strategic Thinking
MP.3 Logic
23
Lesson 18.3
m∠D ∠D ≅ ∠K. So, m∠D = m∠K. 20x + 12 = 25x - 8 → 4 = x
So, m∠D = (20x + 12)° = (20 ⋅ 4 + 12)° = 92°.
So, FG = 2y + 3 = 2(16) + 3 = 32 + 3 = 35 in.
△ GHJ ≅ △ STU by the Transitive Prop.
915
L
(y + 9) in.
19/04/14 11:24 AM
Write each proof.
AVOID COMMON ERRORS
14. Given: Quadrilateral PQTU ≅ quadrilateral QRST _ _ Prove: QT bisects PR.
P
L
Q
R
T
Students may find the value of a variable or the value of an algebraic expression as the solution to a problem when they are in fact only part of the way through the solving process. Remind students to always go back to the initial question to make sure the answer is the solution to the problem.
S
Statements
Reasons
1. Quadrilateral PQTU ≅ quadrilateral QRST
1. Given
2. ¯ PQ ≅ ¯ QR
2. Corr. parts of ≅ fig. are ≅
3. Q is the midpoint of ¯ PR.
3. Definition of midpoint
4. ¯ QT bisects ¯ PR.
4. Definition of segment bisector
15. Given: △ ABC ≅ △ ADC _ _ Prove: AC bisects ∠BAD and AC bisects ∠BCD.
B
A
C
D
Statements
Reasons
1. △ABC ≅ △DEF
1. Given
2. ∠BAC ≅ ∠DAC
2. Corr. parts of ≅ fig. are ≅
3. ∠BCA ≅ ∠DCA
3. Corr. parts of ≅ fig. are ≅
4. ¯ AC bisects ∠BAD and ¯ AC bisects ∠BCD.
4. Definition of angle bisector © Houghton Mifflin Harcourt Publishing Company
16. Given: Pentagon ABCDE ≅ pentagon FGHJK; ∠D ≅ ∠E
Prove: ∠D ≅ ∠K
B
G
A
C
H
F
E
D
J
K
Statements
Reasons
1. Pentagon ABCDE ≅ pentagon FGHJK
1. Given
2. ∠D ≅ ∠E
2. Given
3. ∠E ≅ ∠K
3. Corr. parts of ≅ fig. are ≅
4. ∠D ≅ ∠K
4. Transitive Property of Congruence
Module 18
IN1_MNLESE389762_U7M18L3 916
916
Lesson 3
19/04/14 11:24 AM
Corresponding Parts of Congruent Figures are Congruent
916
△GHJ ≅ △PQR and △PQR ≅ △STU. Complete the following using a side or angle of △STU. Justify your answers.
AVOID COMMON ERRORS Students may write incorrect congruence statements. Make sure they understand that the order of the vertices in a congruence statement is not random. They should know that they can identify corresponding angles by choosing pairs of letters in corresponding positions in a congruence statement. For example, in 4 JZQ ≅ 4 MDH, the letters J and M both appear in the first position in the names of their respective triangles. This means ∠J ≅ ∠M. In a similar way, pairs of letters that are in corresponding positions yield pairs of corresponding sides.
△ABC ≅ △DEF. Find the given side length or angle measure. B
J
E
62° A
71°
C
D
F
M
17. m∠D m∠JAB + m∠BAC = 90°, so 62° + m∠BAC = 90° and m∠BAC = 28°.
Since △ABC ≅ △DEF, ∠BAC ≅ ∠D. m∠BAC = m∠D, and m∠BAC = 28°, so m∠D = 28°.
18. m∠C m∠EFM + m∠EFD = 180°, so 71° + m∠EFD = 180° and m∠EFD = 109°.
Since △ABC ≅ △DEF, ∠C ≅ ∠EFD. m∠C = m∠EFD, and m∠EFD = 109°, so m∠C = 109°.
19. The figure _ the dimensions of two city parks, where △ RST ≅ △ XYZ _ shows and YX ≅ YZ. A city employee wants to order new fences to surround both parks. What is the total length of the fences required to surround the parks? R X 210 ft T Z
320 ft
S
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ken Brown/E+/Getty Images
Y
Since △RST ≅ △XYZ, ¯ ST ≅ ¯ YZ, so ST = YZ = 320 ft. Since ¯ YX ≅ ¯ YZ, YX = YZ = 320 ft. Since the triangles are congruent, they have the same perimeter, which is 210 + 320 + 320 = 850 ft. The total length of the fences is 850 + 850 = 1700 ft. 20. A tower crane is used to lift steel, concrete, and building materials at construction sites. The figure shows part of the horizontal beam of a tower crane, in which △ABG ≅ △BCH ≅ △HGB G
H
59° A
27° B
C
a. Is it possible to determine m∠GBH ? If so, how? If not, why not?
Yes; since corr. parts are ≅, m∠ABG = 27° and m∠HBC = 59°, so
m∠GBH = 180° -59° -27° = 94°.
_ _ b. A member of the construction crew claims that AC is twice as long as AB. Do you agree? Explain. _ _ _ Yes; since corr. parts are ≅, AB ≅ BC and so B is the midpoint of AC .
This means AC is twice AB. Module 18
IN1_MNLESE389762_U7M18L3 917
917
Lesson 18.3
917
Lesson 3
19/04/14 11:24 AM
21. Multi-Step A company installs triangular pools at hotels. All of the pools are congruent and △JKL ≅ △MNP in the figure. What is the perimeter of each pool? J
M
41 ft
(4x - 4) ft
P
L K
INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 When students solve algebraic equations to
(5x + 1) ft
N
(20x - 15)°
find the measures of congruent corresponding parts of figures, caution them to first verify that the correspondences are correct. Suggest that students start by listing the pairs of corresponding parts.
(15x + 15)°
Since corresponding parts are congruent, ∠K ≅ ∠N and so m∠K = m∠N.
20x - 15 = 15x + 15 → x = 6; JK = 4(6) - 4 = 20 ft, KL = 5(6) + 1 = 31 ft
The perimeter of △JKL is 20 + 31 + 41 = 92 ft. The perimeter of △MNP is also 92 ft. 22. Kendall and Ava lay out the course shown below for their radio-controlled trucks. In the figure, △ABD ≅ △CBD. The trucks travel at a constant speed of 15 feet per second. How long does it take a truck to travel on the course from A to B to C to D? Round to the nearest tenth of a second.
PEERTOPEER DISCUSSION Ask students to discuss with a partner how to determine whether two figures are congruent. Have students give each other a pair of figures, look for the congruent corresponding parts, and then write a congruence statement for the figures. Repeat the exercise for other pairs of figures.
B 50 ft
30 ft A
D
40 ft
C
AB + BC + CD = 50 + 50 + 40 = 140 ft; distance = rate × time, so 140 = 15t → t ≈ 9.3 s.
23. △MNP ≅ △QRS. Determine whether each statement about the triangles is true or false. Select the correct answer for each lettered part. M 52 mm
S
52 mm
N
Q
34 mm R
P (2x - 33)°
a. △QRS is isosceles. _ ¯. b. MP is longer than MN
(x + 19)°
True
False
True
False
True
False
d. The perimeter of △QRS is 120 mm.
True
False
e. ∠M ≅ ∠Q
True
False
c.
m∠P = 52°
Module 18
IN1_MNLESE389762_U7M18L3 918
918
a. Since △MNP ≅ △QRS, ¯ MN ≅ ¯ QR, so QR = 52 mm. △QRS has two sides with the same length, so it is isosceles. MP ≅ ¯ QS, b. Since △MNP ≅ △QRS, ¯ so MP = 52 mm. Therefore, MP = MN.
c. Since △MNP ≅ △QRS, ∠P ≅ ∠S, so 2x - 33 = x + 19. Solving the equation shows that x = 52 and m∠P = (2x - 33)° = (2 ⋅ 52 - 33)° = 71°. d. Since △MNP ≅ △QRS, ¯ MP ≅ ¯ QR,
so QR = 52 mm. The perimeter of △QRS is 52 + 52 + 34 = 138 mm.
© Houghton Mifflin Harcourt Publishing Company• Image Credits: ©(T)Image Collective/Alamy, (B) ©Oleksiy Maksymenko Photography/Alamy
Since △ABD ≅ △CBD, ¯ AB ≅ ¯ CB, so AB = 50 ft. Also, ¯ AD ≅ ¯ CD, so CD = 40 ft.
e. Since △MNP ≅ △QRS, ∠M ≅ ∠Q since corresponding parts of congruent figures are congruent.
Lesson 3
19/04/14 11:24 AM
Corresponding Parts of Congruent Figures are Congruent
918
JOURNAL
H.O.T. Focus on Higher Order Thinking
24. Justify Reasoning Given that △ABC _ ≅ △DEF, AB = 2.7 ft, and AC = 3.4 ft, is it possible to determine the length of EF? If so, find the length and justify your steps. If not, explain why not.
Have students write a journal entry in which they state the Corresponding Parts of Congruent Figures Are Congruent Theorem in their own words. Encourage them to include one or more labeled figures as part of the journal entry.
No; the side of △ABC that corresponds to ¯ EF is ¯ BC . The length of this side is not known and cannot be determined from the given information. 25. Explain the Error A student was told that △GHJ ≅ △RST and was asked to find GH. The student’s work is shown below. Explain the error and find the correct answer.
G
S (4x + 3) m
J
5x - 2 = 6x - 5
(6x - 5) m
H
Student's Work -2 = x - 5
T
(5x - 2) m
R
The student incorrectly identified corresponding
3=x
GH = 5x - 2 = 5(3) - 2 = 13 m
sides. Since △GHJ ≅ △RST, ¯ GH ≅ ¯ RS. 5x - 2 = 4x + 3 → x = 5; GH = 5(5) - 2 = 23 m.
26. Critical Thinking In △ABC, m∠A = 55°, m∠B = 50°, and m∠C = 75°. In △DEF, m∠E = 50°, and m∠F = 65°. Is it possible for the triangles to be congruent? Explain.
© Houghton Mifflin Harcourt Publishing Company
No; if the triangles were congruent, then corresponding angles would be congruent. Since m∠F = 65°, there is no angle of △ABC that could be the corresponding angle to ∠F, so the triangles cannot be congruent. _ _ 27. Analyze Relationships △PQR ≅ △SQR and_ RS ≅ RT. A student said that point R appears to be the midpoint of PT. Is it possible to prove this? If so, write the proof. If not, explain why not. Yes;
Statements 1. △PQR ≅ △SQR
2. ¯ RP ≅ ¯ RS 3. ¯ RS ≅ ¯ RT
4. ¯ RP ≅ ¯ RT
5. R is the midpoint of ¯ PT
Module 18
IN1_MNLESE389762_U7M18L3 919
919
Lesson 18.3
P
R
Q
Reasons 1. Given 2. Corr parts of ≅ figs. are ≅
S
T
3. Given 4. Transitive Property 5. Definition of midpoint
919
Lesson 3
19/04/14 11:23 AM
Lesson Performance Task
INTEGRATE MATHEMATICAL PRACTICES Focus on Patterns MP.8 Sketch and number the eight inner triangles
The illustration shows a “Yankee Puzzle” quilt. A
of the Yankee Puzzle quilt on the board. B
1
2
8 C
3 4
7
D
6
a. Use the idea of congruent shapes to describe the design of the quilt.
5
The design is created from 16 congruent triangles. Each quarter of the design consists of 4 of the triangles joined to form a square.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ken Brown/E+/Getty Images
_ b. Explain how the triangle with base AB_ can be transformed to the position of the triangle with base CD.
There are many ways to transform the triangle with base ¯ AB to the position of the triangle with base ¯ CD. One way is to translate it to the position of the triangle directly beneath it, then, rotate it 90° counterclockwise about C, then translate to the right. c. Explain how you know that CD = AB.
CD = AB because corresponding parts of congruent figures are congruent.
Module 18
920
Given Triangle 1, how could you find the locations of the other seven triangles using transformations? Sample answer: Rotate Triangle 1 90°, 180°, and 270° clockwise around the center point to locate triangles 3, 5, and 7. Reflect Triangle 1 across the vertical center line to locate Triangle 2. Then rotate Triangle 2 90°, 180°, and 270° clockwise around the center point to locate triangles 4, 6, and 8.
INTEGRATE MATHEMATICAL PRACTICES Focus on Communication MP.3 Describe how, starting with a square, you could draw the pattern of a Yankee Puzzle quilt. Sample answer: Draw the diagonals of the square. Find the midpoints of the four sides. Connect the midpoint of each side with the midpoint of the side adjacent to it and the midpoint of the side opposite it.
Lesson 3
EXTENSION ACTIVITY IN1_MNLESE389762_U7M18L3 920
Challenge students to draw and color a design for a quilt that meets the following requirements:
19/04/14 11:23 AM
• The design should be square. • The design should consist of triangles and/or quadrilaterals only. • The design should have 90-degree rotational symmetry. Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem.
Corresponding Parts of Congruent Figures are Congruent
920
MODULE
18
Study Guide Review ASSESSMENT AND INTERVENTION
MODULE
STUDY GUIDE REVIEW
18
Congruent Figures Essential Question: How can you use congruency to solve real-world problems?
KEY EXAMPLE
(Lesson 18.1)
Write the vertices of the image of the figure given by A (2, 1), B (3, 3), C (2, 4) after the transformations.
(x, y) → (x + 1, y + 2) → (3x, y). A (2, 1) → A' (3, 3)
Assign or customize module reviews.
MODULE PERFORMANCE TASK
B (3, 3) → B' (4, 5)
Apply the transformations in order to each point. Apply the first transformation.
A' (3, 3) → A" (9, 3)
Apply the second transformation.
C (2, 4) → C' (3, 6)
B' (4, 5) → B" (12, 5) C' (3, 6) → C" (9, 6)
The image of the transformed figure is determined by the points A" (9, 3), B" (12, 5), C" (9, 6).
COMMON CORE
Mathematical Practices: MP.1, MP.3, MP.4, MP.6 G-CO.A.5, G-CO.B.6
SUPPORTING STUDENT REASONING Students should begin this problem by focusing on the transformations needed to move one figure to a congruent figure in the plane. Here are some issues they might bring up. • How to identify an open space on the puzzle that is congruent to one of the available puzzle pieces: The open space on the puzzle has at least two sides that can be matched to the available puzzle pieces. • The rotation(s) needed to position the available puzzle piece: The center of the rotation will be close to the center of the available puzzle piece. • The translation(s) needed to move the available puzzle piece into position: If the rotation to a vertical position is done first, then the translations will be vertical and horizontal. • If there are pieces that will not fit anywhere into the pieces already assembled: Yes, the piece at the upper right appears not to fit anywhere.
921
Module 18
KEY EXAMPLE
(Lesson 18.2)
Determine whether a triangle △ABC is congruent to its image after the transformations (x, y) → (x + 1, y + 2) → (2x, y).
The transformation (x, y) → (x + 1, y + 2) is a translation, which is a rigid motion, so after this transformation the image is congruent. The transformation (x, y) → (2x, y) is a dilation, which is not a rigid motion, so the image from this transformation is not congruent. After the transformations, the image is not congruent to △ABC because one of the transformations is not a rigid motion. KEY EXAMPLE
(Lesson 18.3)
Find the angle in △DFE congruent to ∠A and the side congruent to ¯ BC when △ABC ≅ △DFE. Since_ △ABC _≅ △DFE, and corresponding parts of congruent figures are congruent, ∠A ≅ ∠D and BC ≅ FE.
Module 18
921
Study Guide Review
SCAFFOLDING SUPPORT
IN1_MNLESE389762_U7M18MC 921
• Show students how to use a protractor to approximate the angle of rotation of a puzzle piece. Remind students to specify whether the rotation is clockwise or counterclockwise. • Suggest that students draw a rough sketch of the pieces that have not yet been fitted into the puzzle and number them. Have them also draw a rough sketch of the edge of the puzzle showing the empty spaces and letter the spaces. Then students can begin their descriptions with phrases such as “To move piece 2 into slot E...”.
19/04/14 10:53 AM
EXERCISES Write the vertices of the image of the figure after the transformations. (Lesson 18.1) 1.
SAMPLE SOLUTION
The figure given by A(1, -2), B(2, 5), C(-3, 7), and the transformations A' (3, 2), B' (-4, 4), C' (-6, -6) . x, y → (x, y - 1) → (-y, 2x)
(
I made a sketch of the pieces that had not yet been fitted into the puzzle and numbered them from 1 to 5. I also made a sketch of the empty spaces at the edge of the puzzle and lettered them A to F. Here are the moves I used to transfer the three pieces into the puzzle:
)
Find the rigid motions to transform one figure into its congruent figure. (Lesson 18.2) F
2. In the figure, △ABC ≅ △DEF.
The rigid motions to transform from △ABC ≅ △DEF are (x, y) → (-y, x) → (x -2, y +2) .
Find the congruent parts. (Lesson 18.3) 3. Given △ABC ≅ △DEF, ∠A ≅
∠D .
_ 4. Given △ABC ≅ △DEF, CA ≅
¯ FD .
E D -8
-4
8
y B
4 A
C x
0 -4
4
To move Piece 4 into Space C, rotate the piece approximately 75 degrees clockwise, then move it vertically upward and horizontally to the left.
8
-8
To move Piece 1 into Space D, rotate the piece approximately 150 degrees counterclockwise, then move it vertically downward and horizontally to the left. To move Piece 2 into Space A, rotate the piece approximately 100 degrees counterclockwise, then move it vertically upward and horizontally to the left.
MODULE PERFORMANCE TASK
Jigsaw Puzzle A popular pastime, jigsaw puzzles are analogous to the series of transformations that can be performed to move one figure onto another congruent figure. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Hitdelight/Shutterstock
In the photo, identify at least three pieces that would likely fit into one of the empty spaces in the puzzle. Describe the rotations and translations necessary to move the piece to its correct position in the puzzle.
Module 18
922
Study Guide Review
DISCUSSION OPPORTUNITIES
IN1_MNLESE389762_U7M18MC 922
19/04/14 10:53 AM
• Can any available piece be moved into position using a single transformation? • The available pieces look very similar. How will you determine whether you chose the correct piece to transform to the congruent piece?
Assessment Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain. 0 points: Student does not demonstrate understanding of the problem.
Study Guide Review 922
Ready to Go On?
Ready to Go On?
18.1–18.3 Congruent Figures
ASSESS MASTERY Use the assessment on this page to determine if students have mastered the concepts and standards covered in this module.
• Online Homework • Hints and Help • Extra Practice
Predict the results of the transformations. (Lesson 18.1)
1. Triangle △ABC is in the first quadrant and translated along ⟨2, 1⟩ and reflected across the x-axis. The first quadrant Which quadrant will the triangle be in after the first transformation? Which quadrant will the triangle be in after the second transformation? The fourth quadrant
ASSESSMENT AND INTERVENTION
Determine whether the triangles are congruent using rigid motions. (Lesson 18.2) 2. Using the graph with △ABC, △DEF, and △PQR:
A. Determine whether △ABC is congruent to △DEF. △ABC is not congruent to △DEF.
Access Ready to Go On? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.
Differentiated Instruction Resources • Reading Strategies • Success for English Learners • Challenge Worksheets Assessment Resources
R
D -2 P
B
2 0 -2
A 2
Cx 4
Q
△DEF is congruent to △PQR. Find the congruent parts of the triangles. (Lesson 18.3) 3. List all of the pairs of congruent sides for two congruent triangles △ABC and △DEF.
¯ AB ≅ ¯ DE , ¯ BC ≅ ¯ EF , ¯ CA ≅ ¯ FD
© Houghton Mifflin Harcourt Publishing Company
• Reteach Worksheets
E -4
B. Determine whether △DEF is congruent to △PQR.
ADDITIONAL RESOURCES Response to Intervention Resources
F 4 y
ESSENTIAL QUESTION 4. How can you determine whether a shape is congruent to another shape?
Answers may vary. Sample: You can figure out whether a shape is congruent to another by determining whether a sequence of rigid motions will transform one shape into the other.
• Leveled Module Quizzes
Module 18
COMMON CORE IN1_MNLESE389762_U7M18MC 923
923
Module 18
Study Guide Review
923
Common Core Standards
19/04/14 10:53 AM
Content Standards Mathematical Practices
Lesson
Items
17.1
1
G-CO.A.5
MP.4
17.1, 17.2
2
G-CO.B.7
MP.7
17.3
3
G-CO.B.7
MP.4
MODULE MODULE 18 MIXED REVIEW
MIXED REVIEW
Assessment Readiness
Assessment Readiness
1. A line segment with points R(3, 5) and S(5, 5) is reflected across the line y = -x and translated 2 units down. Determine whether each choice is a coordinate of the image of the line segment. Select Yes or No for A–C. A. R' (-5, -3) Yes No B. R' (-5, -5) C. S' (-5, -7)
18
Yes Yes
ASSESSMENT AND INTERVENTION
No No
2. The polygon ABCD is congruent to PQRS. The measure of angle B is equal to 65°. Choose True or False for each statement. A. The supplement of angle Q measures 115°.
True
False
B. Angle Q measures 115°.
True
False
C. The supplement of angle B measures 115°.
True
False
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
ADDITIONAL RESOURCES
3. Triangle LMN is a right triangle. The measure of angle LML is equal to 35°. Triangle LMN is congruent to △PRQ with right angle R. Choose True or False for each statement. A. The measure of angle Q is 55°.
True
False
B. The measure of angle R is 90°.
True
False
C. The measure of angle P is 35°.
True
False
¯; ED ¯ FD
Module 18
COMMON CORE
• Leveled Module Quizzes: Modified, B
AVOID COMMON ERRORS Item 1 Some students will stop too soon when faced with a problem with multiple steps. Encourage students to number each step, and then make sure they have completed each one before choosing a final answer to the problem.
© Houghton Mifflin Harcourt Publishing Company
_ 4. The two triangles, △ABC and _△DEF, are congruent. Which side is congruent to CA? Which side is congruent to BA?
Assessment Resources
Study Guide Review
924
Common Core Standards
IN1_MNLESE389762_U7M18MC 924
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Content Standards Mathematical Practices
Lesson
Items
17.1, 16.1, 16.2
1*
G-CO.A.5
MP.5
17.3, 15.2
2*
G-CO.A.1, G-CO.B.6
MP.2
17.3
3
G-CO.B.6
MP.2
17.3
4
G-CO.B.6
MP.7
* Item integrates mixed review concepts from previous modules or a previous course.
Study Guide Review 924
UNIT
7
UNIT 7 MIXED REVIEW
Assessment Readiness
MIXED REVIEW
Assessment Readiness
1. Consider each expression. if x = -2, is the value of the expression a positive number? Select Yes or No.
ASSESSMENT AND INTERVENTION
A. -2(x - 2)
2
Yes Yes Yes
B. -3x(5 - 4x) C. x 3 + 6x
2. A bedroom is shaped like a rectangular prism. The floor has a length of 4.57 meters and a width of 4.04 meters. The height of the room is 2.3 meters. Choose True or False for each statement. A. The perimeter of the floor with the correct number of significant digits is 17.22 meters. True False B. The area of the floor with the correct number of significant digits is 18.46 square meters. True False C. The volume of the room with the correct number of significant digits is 42 cubic meters. True False
Assign ready-made or customized practice tests to prepare students for high-stakes tests.
ADDITIONAL RESOURCES Assessment Resources • Leveled Unit Tests: Modified, A, B, C
3. Does the ray BD bisect ∠ABC? Select Yes or No for each pair of angles. A. m∠ABC = 60°, m∠ABD = 30° B. m∠ABC = 96°, m∠ABD = 47°
• Performance Assessment
AVOID COMMON ERRORS
C. m∠ABC = 124°, m∠ABD = 62°
© Houghton Mifflin Harcourt Publishing Company
Item 7 Some students will attempt this problem without plotting the transformations. Encourage students to use a sheet of graph paper and test each transformation.
No No No
• Online Homework • Hints and Help • Extra Practice
B. R(-1, 3), S(2, -2), and D(-4, 2), F(-1, -3) C. R(5, -3), S(2, 2), and D(1, -4), F(-1, -3)
Unit 7
COMMON CORE
Yes
No
Yes Yes Yes
No No No
Yes
No
Yes Yes
No No
―
5. Is RS a translation of DF? Select Yes or No for each statement. A. R(2, 2), S(5, 2), and D(3, 3), F(5, 3)
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No No
―
4. Is the point C the midpoint of the line AB? Select Yes or No for each statement. A. A(1, 2), B(3, 4), and C(2, 3) B. A(-1, 2), B(3, -1), and C(1, 0) C. A(-3, 0), B(-1, 5), and C(-2, 2)
―
Yes Yes
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Common Core Standards Content Standards
Mathematical Practices
1
A-REI.B.3
MP.2
2
G-GMD.B.4
MP.6
3
G-CO.D.12
MP.6
4
G-CO.A.1
MP.6
5
N-CN.B.6
MP.5
Items
* Item integrates mixed review concepts from previous modules or a previous course.
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6. Does the shape have rotational symmetry? Select Yes or No for each statement. A. A square B. A trapezoid C. A right triangle
Yes Yes Yes
PERFORMANCE TASKS
No No No
There are three different levels of performance tasks: * Novice: These are short word problems that require students to apply the math they have learned in straightforward, real-world situations.
7. Determine whether each image of △ABC, with A(1, 3), B(2, 3), C(4, 5), can be formed with only the given transformation. Select True or False for each statement. A. A′(2, 4), B′(3, 4), C′(5, 6) is formed by translation. True False B. A′(-1, 3), B′(-2, 3), C′(-4, 5) is formed by rotation. True False C. A′(1, -5), B′(2, -3), C′(4, -1) is formed by reflection. True False
** Apprentice: These are more involved problems that guide students step-by-step through more complex tasks. These exercises include more complicated reasoning, writing, and open ended elements.
8. For △DEF, with D(2, 2), E(3, 5), F(4, 3), and △D′E′F′, with D′(4, 2), E′(3, 5), F′(2, 3), determine whether the image can be formed with the sequence of transformations. Select True or False for each statement. A. The image is formed by a reflection followed by a translation. True False B. The image is formed by a rotation followed by a reflection. True False C. The image is formed by two consecutive reflections. True False 9. Use the figure to answer the questions below. A. What is a specific series of rigid transformations that maps △ABC onto △DEF?
B. List all congruent pairs of angles and sides for the two figures.
――
―
――
―
y
2 -4
A 0 -2
B C
F E D 2
x 4
-4
AB ≅ DE, BC ≅ EF, CA ≅ FD
∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F
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© Houghton Mifflin Harcourt Publishing Company
Answers may vary. Sample: Reflect across the x-axis and translate to the right 5 units and up 1 unit.
4
***Expert: These are open-ended, nonroutine problems that, instead of stepping the students through, ask them to choose their own methods for solving and justify their answers and reasoning.
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COMMON CORE IN1_MNLESE389762_U7UC 926
Common Core Standards Content Standards
Mathematical Practices
6
G-CO.A.2
MP.5
7
G-CO.A.3
MP.5
8
G-CO.A.5
MP.5
9
G-CO.A.5, G-CO.A.6
MP.2, MP.7
Items
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* Item integrates mixed review concepts from previous modules or a previous course.
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Performance Tasks
SCORING GUIDES
10. A student has drawn a figure of a square PQRS with points P(-5, 5), Q(1, 5), R(1, -1), and S(-5, -1). For the next assignment, the teacher wants students to ― inscribe another square, but with sides of length √18 , in the square. How would a student find the correct square? What are the vertices of the inscribed square?
Item 10 (2 points) Award the student 1 point for a correct explanation of how to find the square, and 1 point for the correct vertices (–2, 5), (1, 2), (–2, –1), and (–5, 2).
Note that the square’s side lengths are 6. All sides must be the same length, so the midpoints of each square side should be found. Confirm that using the midpoints of the square as the vertices for the inscribed square gives a ― square with side length √18 . The vertices are (-2, 5), (1, 2), (-2, -1), and (-5, 2).
Item 11 (6 points) 2 points for naming correct transformation type
11. A square table is set with four identical place settings, one on each side of the table. Each setting consists of a plate and spoon. Choose one as the original place setting. What transformation describes the location of each of the other three? Express your answer in terms of degrees, lines of reflection, or directions from the original place setting.
4 points for full description Item 12 (6 points) A. 1 point for correct answer about straight lines
Possible answer: They are rotations of the first place setting with the center of rotation in the center of the table. The second setting is a rotation of 90 degrees, the third is 180 degrees, and the fourth is 270 degrees.
1 point for correct answer about nonintersecting lines B. 1 point for correct answer about straight lines 1 point for correct answer about nonintersecting lines
12. In spherical geometry, the plane is replaced by the surface of a sphere. In this context, straight lines are defined as great circles, which are circles that have the same center as the sphere. They are the largest possible circles on the surface of the sphere.
C. 2 points for correct answer
© Houghton Mifflin Harcourt Publishing Company
A. On a globe, lines of longitude run north and south. In spherical geometry, are lines of longitude straight lines? Are any lines of longitude parallel (nonintersecting)? B. Lines of latitude run east and west. In spherical geometry, are lines of latitude straight lines? Are any lines of latitude parallel (nonintersecting)? C. In general, in how many places does a pair of straight lines intersect in spherical geometry? A. Lines of longitude are straight lines, and no lines of longitude are nonintersecting. B. Most lines of latitude are not straight lines, but the equator is straight. All lines of latitude are nonintersecting. C. All straight lines in spherical geometry intersect in exactly two places.
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math in careers
MATH IN CAREERS
Geomatics surveyor A geomatics surveyor is surveying a piece of land of length 400 feet and width 300 feet. Standing at one corner, he finds that the elevation of the opposite corner is 50 feet greater than his elevation. Find the distance between the surveyor and the middlemost point of the piece of land (ignoring elevation), the elevation of the middlemost point in comparison to his location (assuming that the elevation increases at a constant rate), and distance between the surveyor and the middlemost point of the piece of land considering its elevation.
Geomatics Surveyor In this Unit Performance Task, students can see how a geomatics surveyor uses mathematics on the job. For more information about careers in mathematics as well as various mathematics appreciation topics, visit the American Mathematical Society http://www.ams.org
The distance, ignoring elevation, is found with the distance formula.
――――― 300 400
√(___) + (___) = 250 feet 2
2
2
2
The elevation is found by dividing by 2 feet.
――――― ≈ 251.25 feet
The final distance is found using the distance formula.
√(250) 2 + (25) 2
SCORING GUIDES Task (6 points) 2 points for the correct distance ignoring elevation 2 points for finding elevation 2 points for correct distance including elevation
© Houghton Mifflin Harcourt Publishing Company
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